Abstract
Understanding how heterogeneous neural populations represent sensory input to give rise to behavior remains a central problem in systems neuroscience. Here we investigated how midbrain neurons within the electrosensory system of Apteronotus leptorhynchus code for object location in space. In vivo simultaneous recordings were achieved via Neuropixels probes, high-density electrode arrays, with the stimulus positioned at different locations relative to the animal. Midbrain neurons exhibited heterogeneous response profiles, with a significant proportion (65%) seemingly nonresponsive to moving stimuli. Remarkably, we found that nonresponsive neurons increased population coding of object location through synergistic interactions with responsive neurons by effectively reducing noise. Mathematical modeling demonstrated that increased response heterogeneity together with the experimentally observed correlations was sufficient to give rise to independent encoding by responsive neurons. Furthermore, the addition of nonresponsive neurons in the model gave rise to synergistic population coding. Taken together, our findings reveal that nonresponsive neurons, which are frequently excluded from analysis, can significantly improve population coding of object location through synergistic interactions with responsive neurons. Combinations of responsive and nonresponsive neurons have been observed in sensory systems across taxa; it is likely that similar synergistic interactions improve population coding across modalities and behavioral tasks.
Significance Statement
Here we show that including the activities of nonresponsive neurons with those of responsive neurons increases Fisher information about stimulus location. Further analysis revealed that this is because including nonresponsive neurons led to reduced noise levels for responsive neurons. A combination of multiunit recordings from neural populations and mathematical modeling reveals that response heterogeneity and spatially decaying correlations are necessary to observe this effect. It is likely that synergistic population coding by responsive and nonresponsive neurons will be observed in other systems.
Introduction
Investigating how the activities of neural populations are combined to give rise to behavior (i.e., population coding) remains an important area of study in neuroscience. Population coding has been studied extensively, both experimentally and theoretically (Averbeck et al., 2006; Averbeck and Lee, 2006; Moreno-Bote et al., 2014; Lin et al., 2015; Franke et al., 2016; Kohn et al., 2016; Zylberberg et al., 2016; Perez-Nieves et al., 2021; Panzeri et al., 2022; Urai et al., 2022). Of key importance is the fact that neural activities are correlated with one another (deCharms and Merzenich, 1996; Usrey and Reid, 1999; Chacron and Bastian, 2008; Cohen and Maunsell, 2009; Cohen and Kohn, 2011; Hong et al., 2012; Doiron et al., 2016). In particular, there is much debate as to how noise correlations (i.e., correlations between the response trial-to-trial variabilities), which should be distinguished from signal correlations (i.e., correlations between the average neural responses), influence information transmission (Ecker et al., 2011; Kanitscheider et al., 2015; Franke et al., 2016; Panzeri et al., 2022; Urai et al., 2022; Kira et al., 2023).
However, previous studies have not considered the impact of neurons that do not respond to a given stimulus (i.e., are nonresponsive) on population coding, despite the fact that these are frequently encountered experimentally (Pasupathy and Connor, 2002; Fiscella et al., 2015; Franke et al., 2016; Nigam et al., 2019; Ni et al., 2022). It is commonly assumed that, because nonresponsive neurons are not tuned to stimulus attributes, they cannot contribute to information transmission. While valid at the individual neuron level, this assumption is in general not correct at the population level. This is because nonresponsive neurons can interact with responsive neurons and therefore influence information transmission by shaping noise correlations.
Gymnotiform wave-type weakly electric fish offer an attractive model for studying population coding because the anatomy (Maler et al., 1991) and physiology (Bell and Maler, 2005; Chacron et al., 2011; Krahe and Maler, 2014; Metzen and Chacron, 2019) of the electrosensory system has been well characterized. These fish actively generate a quasisinusoidal electric field, the electric organ discharge (EOD), around their body and can sense perturbations of this field caused by objects in the environment. Amplitude modulations in the EOD are encoded by electroreceptor afferents (EAs) embedded in the epidermis (Fortune, 2006) whose afferents synapse onto electrosensory lateral line lobe (ELL) pyramidal cells within the hindbrain ELL (Turner et al., 1999). ELL pyramidal cells project to the midbrain torus semicircularis (TS), which is homologous to the mammalian inferior colliculus. TS neurons project to higher brain areas such as the preglomerular nucleus (PG), tectum, and nucleus electrosensorius (Carr et al., 1981; Carr and Maler, 1985; Maler et al., 1991; Bell and Maler, 2005; Krahe and Maler, 2014).
Behavioral studies have shown that these fish are adept at localizing and capturing prey despite the faint electric signals (Nelson and Maciver, 1999; Nelson et al., 2002; Babineau et al., 2007; Snyder et al., 2007). Previous studies have focused on the neural mechanisms mediating these extraordinary behavioral abilities at multiple levels. At the sensory periphery, it was found that the action potential patterns of individual EAs allows for reduced variability at the timescale associated with prey detection (Chacron et al., 2001; Brandman and Nelson, 2002; Nesse et al., 2010, 2021). While EAs do not display noise correlations (Metzen et al., 2015, 2016), this is not the case for their downstream ELL pyramidal cell targets (Chacron and Bastian, 2008). Such noise correlations were not initially taken into account (Jung et al., 2016), most likely because it was difficult to obtain recordings from multiple (i.e., >3) electrosensory neurons simultaneously.
However, technological advances such as Neuropixels probes (Jun et al., 2017) have made simultaneous recording of large ELL neural populations possible and thus allowed investigation as to how noise correlations impact population coding (Wang and Chacron, 2021; Haggard and Chacron, 2023; Marquez and Chacron, 2023; Metzen and Chacron, 2023). In the case of population coding of object location, it was found that noise correlations in ELL pyramidal cells introduce redundancy, thereby limiting information transmission (Haggard and Chacron, 2023). However, how such information is decoded downstream has not been investigated to date. Here we recorded from TS neural populations and demonstrate that nonresponsive neurons increase information about object location by reducing noise.
Materials and Methods
Ethics statement
Animal care and experimental procedures were approved by the McGill University Animal Care Committee (protocol number 5285) and adhered to the Canadian Council on Animal Care guidelines.
Animal care
In this investigation, we exclusively used specimens of the wave-type weakly electric fish, Apteronotus leptorhynchus (N = 10, five for recordings from ELL and five for recordings from TS; sex unknown). The fish were procured from tropical fish dealers and were subsequently housed in tanks with groupings of up to 10 fish per tank. The environmental conditions in these tanks were maintained within prescribed parameters, including a temperature range of 26–29°C and a conductivity range of 100–800 µS/cm, in accordance with published guidelines for this species (Hitschfeld et al., 2009).
Surgery
The surgical techniques employed in our study have been comprehensively documented in previous research (Bastian et al., 2002; Chacron and Bastian, 2008; Metzen and Chacron, 2021, 2023; Wang and Chacron, 2021; Haggard and Chacron, 2023; Marquez and Chacron, 2023; Vazquez-Guerrero et al., 2024). In summary, we filled an experimental tank (30 × 30 × 10 cm) with water sourced from the fish's home tank. The water was heated and oxygenated in a separate reservoir and was continuously recirculated into the experimental tank. The fish was immobilized using an intramuscular injection of 0.1–0.5 mg of tubocurarine chloride hydrate (Sigma-Aldrich). Following cessation of gilling, the fish was positioned in the experimental tank and received a constant flow of water through a respiration tube inserted into the mouth, with a flow rate of ∼10 ml/min. A 5% topical lidocaine ointment (AstraZeneca) was applied to anesthetize the skin over the skull, a portion of the skin was then removed, and a head post was securely glued to the anterior portion of the skull to ensure stability. A small window of ∼5 mm2 was created, exposing the ipsilateral side of the hindbrain for electrophysiological recording within ELL or exposing the contralateral side of the midbrain for recording within TS. Saline solution was applied regularly to the exposed brain to prevent tissue dehydration.
Recordings and spike sorting
Electrophysiological extracellular recordings of populations of ELL and TS neurons were obtained using Neuropixels 1.0 probes (Imec; Metzen and Chacron, 2021, 2023; Wang and Chacron, 2021; Haggard and Chacron, 2023; Marquez and Chacron, 2023). Anatomical landmarks and physiological responses were used to guide the probe position. The recordings were digitized at 30 kHz using the software package SpikeGLX (Janelia Research Campus, Howard Hughes Medical Institute) and then stored for off-line analysis.
Spike sorting was done using Spike2 (Cambridge Electronic Designs) for the TS data, while ELL neurons were sorted using the automatic spike sorting algorithm Kilsort2 (https://github.com/MouseLand/Kilosort2, developed for electrophysiological data with high-channel counts), followed by manual curation using Phy2 (https://github.com/cortex-lab/phy, a graphical interface for visualization and manual curation of multielectrode data). This is because there are important differences between recordings from TS and ELL. Notably, the spiking activities of TS neurons appeared primarily on a single probe recording site with minimal drift and only a few cases of more than one neuron showing activity on overlapping channels. This is most likely due to the fact TS neurons are small on average. In contrast, ELL neurons are larger with dendrites extending up to 1 mm away from the soma (Bastian et al., 2004). As such, neural activity from a given neuron was typically seen on 5–7 probe sites and frequently overlapped with other units (i.e., the activities of different neurons could be observed on the same recording sites). Furthermore, activity showed significant drift throughout recording. Programs like Kilosort2 were designed to sort neural activities recorded on overlapping sets of probe sites like our ELL data as well as to take drift into account (Pachitariu et al., 2024). However, we found that Kilosort2 gave rise to poor results on TS datasets, which is why we instead used Spike2.
After initial spike sorting, inclusion criteria used to confirm that both ELL and TS neurons were well isolated included recording stability, firing rate, interspike interval, spike waveform, and autocorrelograms (ACGs), adjusted appropriately relative to the brain region of each recording session. The sorted neural activity was then imported into MATLAB (MathWorks) where custom code was used to analyze the data as described below.
The distance between neurons was obtained from the Euclidean distance between the electrode sites on the Neuropixels probe where the neurons in each pair have the highest amplitude spikes, as done previously (Metzen and Chacron, 2021; Wang and Chacron, 2021; Haggard and Chacron, 2023). We quantified the overlap between distributions of distances using an overlapping index (Pastore and Calcagni, 2019) as follows:
The TS dataset contained a total of n = 209 neurons. These neurons were recorded over five experiments, one experiment per fish; during each experiment, we positioned the Neuropixels probe in the TS and recorded the population of neurons in response to the full stimulus protocol. We then repositioned the Neuropixels probe and repeated the stimulus protocol; this allowed us to collect between one and four datasets during each experiment. The neural population sizes for each dataset are as follows: Experiment 1, n = 25 and 18; Experiment 2, n = 27, 17, and 13; Experiment 3, n = 38, 19, 16, and 7; Experiment 4, n = 13; and Experiment 5, n = 16. The ELL dataset contained a total of n = 158 neurons recorded over five experiments, with one dataset collected during each experiment, producing five populations of simultaneously recorded neurons (n = 31, 34, 30, 34, 29, respectively).
Stimulation
Stimulus delivery
A. leptorhynchus possesses a neurogenic EOD. Therefore, the animal continues emitting its EOD after immobilization with tubocurarine. To record the EOD, electrodes were positioned at the rostral and caudal ends of the fish. A sinusoidal waveform was triggered by a function generator (33220 A LXI arbitrary waveform generator, Agilent), when the EOD signal crossed zero from below [121 Window discriminator, World Precision Instruments (WPI)], with a frequency ∼30 Hz higher than that of the EOD so as to remain synchronized with the EOD. The EOD-triggered waveform was then multiplied by the stimulus waveform to generate the desired amplitude modulation (MT3 analog multiplier, Tucker-Davis Technologies). The stimulus was electrically isolated from the ground (A395 Linear Stimulus Isolator, WPI) and delivered to the fish, with a sampling rate of 2 kHz, through two separate sets of electrodes. (1) Spatially diffuse stimuli (i.e., global stimuli) were delivered via two steel wire electrodes that were positioned ∼15 cm lateral on either side of the animal. These consisted of a zero-mean Gaussian white noise stimulus that was low-pass filtered (eighth-order Butterworth, 120 Hz cutoff frequency) with contrasts of 15–25%. (2) Spatially localized stimuli (i.e., local stimuli) were delivered via a small dipole that was positioned perpendicularly to the animal's rostrocaudal axis 1–2 mm away from the skin. The dipole was built from two insulated stainless steel wires (tip spacing ∼2 mm) and was mounted on an actuator that allowed movement along the rostrocaudal axis, as done previously (Bastian et al., 2002; Chacron and Bastian, 2008). We focused on the rostrocaudal axis because this is the primary axis of motion during the prey detection-to-capture behavior sequence and because the receptive fields are elongated in the rostrocaudal axis compared with the dorsoventral axis (Nelson and Maciver, 1999; MacIver et al., 2001; Bastian et al., 2002). The stimulus waveforms were designed such as to replicate the electric images caused by prey (Haggard and Chacron, 2023). Specifically, we used 4 Hz sinusoidal waveforms with contrasts of 15–25% as this frequency corresponds to the timecourse of the electric image at a specific location on the skin as a hunting fish passes by its prey (Nelson and Maciver, 1999).
Stimulation protocol
Prior to initializing the stimulus protocol, the approximate center and edges of the receptive fields of the neurons being recorded was assessed by placing the dipole at different locations and playing the local stimulus while visually assessing the strength of the population activity (e.g., how phase locked is the neural activity to the 4 Hz sinusoidal stimulus). The dipole was then positioned at the dorsoventral location that gave rise to the strongest responses and moved rostrally until neural responses could no longer be elicited. We empirically found that this methodology was sufficient based on subsequent analyses and quantification of spatial receptive fields (see below). Neural activities were first recorded in the absence of stimulation but in the presence of the animal's unmodulated EOD (i.e., “baseline”) for 100 s. The stimulation protocol was then initiated. First, the spatially diffuse stimulus was presented once for 50 s. The 4 Hz sinusoidal stimulus was then presented for a total of 100 trials, with a trial defined as the duration of one full cycle (0.25 s). The dipole was then repositioned caudally at 0.5 cm increments while recording in ELL and 0.25 cm increments while recording in TS, and 100 trials of the 4 Hz sinusoidal stimulus were presented at each location. We used a higher spatial resolution for TS recordings to characterize their activity more accurately. This is because TS neurons are more heterogeneous and have not been characterized as thoroughly as ELL neurons. We collected data at a range of positions in order to map the spatial receptive fields of the recorded neurons along the rostrocaudal axis for a given recording session. The number of positions ranged from 10 to 17 and from 12 to 33 for ELL and TS recordings, respectively.
Data analysis
Data processing
The sorted neural activities were imported into MATLAB as spike times which were then converted to a “binary” time series (1 when a spike occurs and 0 otherwise) with a sampling rate of 2 kHz. This sampling rate was chosen because its inverse (0.5 ms) is less than the refractory period of both TS and ELL neurons (Toporikova and Chacron, 2009; Vonderschen and Chacron, 2011), such that at most one spike can occur over the duration of any given bin. The binary sequences were then converted into spike-count sequences as follows. During spatially localized stimulation, spike counts were obtained during time windows corresponding to the positive half-cycle of the stimulus (0.125 s) for each trial. During baseline activity, spike counts were obtained during nonoverlapping and consecutive time windows of 0.125 s. Alternatively, the binary time series was high-pass filtered (second-order Butterworth filter with a 0.01 Hz cutoff), and the baseline firing rate was calculated by averaging the filtered firing rate. We found that both approaches gave rise to similar values for the baseline firing rate.
Correlations and covariance
Correlations were obtained from the Pearson's correlation coefficient between the spike-count sequences of neural pairs. We also computed the covariance between the spike-count sequences of neural pairs. Correlations and covariances between pairs are reported as matrices in the results. In particular, we shall denote the covariance matrix whose element ij is the covariance between the spike-count sequences of neurons i and j at spatial location x as cov(x).
We note that, in the case of stimulation, the spike counts are obtained during repeated presentations of the same stimulus. Therefore, any variation in the spike count can only be due to trial-to-trial variability of the neural response and not to any variation in the stimulus. As a result, correlations between the spike counts are equal to those obtained from the residual spike counts (i.e., the spike count minus its mean value over trials). The spike-count correlations reported in this study are thus effectively “noise correlations.” We chose to refer to them as spike-count correlations throughout in order to be consistent with the terminology used in previous studies (Panzeri et al., 2022). To determine if spike-count correlations were significantly different from zero, we compared values obtained from the intact data (i.e., “data”) with values obtained after randomly shuffling stimulus trials (i.e., “shuffled”).
To test if the spike-count correlations were spatially modulated, we computed the percentage spatial variation by calculating the median of the covariance magnitude for each location after aligning and pooling the datasets. We then divided the standard deviation (SD) by the mean and multiplied by 100.
Spatial receptive fields
For each neuron, we obtained the spatial receptive field by averaging the spike-count sequence obtained during spatially localized stimulation (i.e., 4 Hz sinusoid) for each rostrocaudal position and divided by the length of the time window (i.e., 0.125 s) to obtain the firing rate (Bastian et al., 2002; Haggard and Chacron, 2023). The firing rate was then plotted as a function of spatial location to obtain the spatial receptive field. We shall denote the spatial receptive field of neuron i as
Baseline firing rates
The pooled baseline firing rates were binned and fitted with an exponential distribution (μ = 8.51; R2 = 0.94; n = 15).
Segregation of ELL neurons into ON- versus OFF-type
We used both the spatial receptive field and the response to the spatially diffuse noise stimulus to classify ELL neurons into either ON- or OFF-type. For the former, we first identified the spatial location where 4 Hz stimulation led to the greatest magnitude difference between the firing rate and the baseline value. The sign of the difference was then used to determine whether each neuron is ON- or OFF-type. Specifically, neurons for which the difference was positive were classified as ON-type, and neurons for which the difference was negative were classified as OFF-type.
Alternatively, ELL neurons were classified as ON- or OFF-type based on responses to the spatially diffuse noise stimulus as done previously (Martinez et al., 2016). We computed the spike-triggered average (STA) by averaging the stimulus waveforms during time windows of 100 ms duration centered on each spike time. The average slope of the STA within a 10 ms time window from −12 to −2 ms before the spike time was then computed. The distribution of slopes was clearly bimodal with all values either being positive or negative (Hartigan's dip test, p = 0.01; dip at slope, 0.08; n = 40). Neurons for which the slope is positive were classified as ON-type, while neurons for which the slope was negative were classified as OFF-type. Overall, we found that both classification schemes led to the same results for our ELL dataset.
Segregation of TS neurons into ON-type, OFF-type, and nonresponsive
Unlike ELL neurons that respond to both spatially localized and spatially diffuse stimulation, TS neurons are in general more selective. Indeed, some TS neurons in our dataset responded to spatially localized but not spatially diffuse stimuli and vice versa. Therefore, we used criteria to segregate TS neurons that were slightly different than those used for the ELL data. Because our study was focused on population encoding of spatial location, we primarily used spatial receptive fields to classify TS neurons.
We used the following criteria to determine whether TS neurons were spatially tuned and thus contribute to object localization. We computed the ACG of the receptive field and obtained the difference between the value at 0.5 cm lag and that at 0 cm lag. To determine whether this value was significantly different from zero, we randomly shuffled the firing rates from the spatial receptive field over location and recomputed the ACG as well as the difference value. We note that this procedure preserves the first-order statistics of the firing rate (i.e., the distribution of the firing rate). This process was repeated 200 times to build a distribution of surrogate data that was approximately normal by the central limit theorem. The 95% confidence interval of this distribution was then computed. If the experimentally observed difference value obtained from the intact data fell outside of the confidence interval, the neuron was determined to be spatially modulated and thus categorized as “responsive.” Neurons for which this was not the case were categorized as “nonresponsive.” We considered pairings between responsive and nonresponsive neurons, between nonresponsive neurons, and between responsive neurons when computing spike-count correlations as well as overlap between distance distributions.
Responsive TS neurons typically displayed spatial receptive fields with antagonistic center-surround organization. Thus, criteria like those used above for ELL neurons based on the spatial receptive field were used to determine whether a neuron was ON- or OFF-type. We found however that there were significant differences between TS and ELL spatial receptive fields. Specifically, the baseline firing rate of many TS neurons was close to zero, leading to a nonlinear rectification of the receptive field in the region where the firing would normally be less than during the baseline. In these cases, the neural activity during the negative half-cycle of the stimulus was used to determine whether this region was part of the receptive center or surround and cells classified as either OFF- or ON-type, respectively.
Fisher information
Definition and calculation
To study how well the fish can determine the location of prey given the recorded neural activity in TS, we computed the Fisher information (FI; Abbott and Dayan, 1999; Moreno-Bote et al., 2014; Franke et al., 2016). FI quantifies the inverse variance of the optimal unbiased estimator of a variable and is a function of the slope of the spatial receptive fields and covariance matrices and is given by the following:
In practice, to reduce the variability due to a finite number of trials, the slope of the receptive fields and covariance matrices were calculated as the average of these variables between two locations as done previously (Kanitscheider et al., 2015). Before inverting the covariance matrices, we set a minimum variance of 0.01 spikes2/trial as done previously (Franke et al., 2016), such that the matrix was not singular. In addition, we increased the sampling of the receptive field, by interpolation, prior to calculating the slope. Because the spatial sampling rate of the TS neural recordings (0.25 cm) was higher than that of the ELL neural recordings (0.5 cm), we used interpolation of both the receptive field and covariance matrices of the ELL data prior to calculating the FI so that we could compare ELL and TS.
Cramér–Rao bound and spatial resolution
To compare the accuracy of the location estimate in TS and ELL with the size of the prey, we converted the FI to the Cramér–Rao bound. Specifically, the Cramér–Rao bound is equal to the inverse FI as follows:
Effects of spike-count correlations on FI
To investigate the effect of spike-count correlations on the FI, we compared the FI of the intact data (with rSC) with the case where the neural activity was independent (without rSC). Independence is typically achieved one of two ways, by artificially setting all of the off-diagonal values of the covariance matrices to zero or by independently shuffling the trials of each neuron in the pair to disrupt the spike-count correlations (as discussed in the previous section). We compared both methods and found similar results. The results reported in the figures were obtained using the first method.
We note that, in the case of an independent population (i.e., all off-diagonal elements of the covariance matrix are zero), the FI for n neurons is simply the sum of the contributions of each individual neuron as follows:
Spatially averaged FI and error calculation
We averaged the FI between locations x = −0.75 cm and x = 0.75 cm to obtain the spatially averaged FI, which included the locations where most of the recorded neurons were maximally informative. We resampled the data, by bootstrapping, to build populations of sizes 1 to n − 1 neurons (where n is the maximum number of neurons in a given dataset). To investigate any differences in the information contained in a population of responsive neurons versus a population of both responsive and nonresponsive neurons, we repeated this analysis but first resampled only the responsive neurons up until one less than the maximum number of responsive neurons and then started adding nonresponsive neurons.
We also wanted to determine how the information encoded in the TS populations differed from that in the ELL. We started by pooling the maximum FI of each individual neuron in the TS and ELL recordings (the maximum FI can occur at different locations for different neurons, even within a given experiment, so this calculation is independent of stimulus location). Next, we calculated the FI as a function of the population size as described above; however because the FI of the TS populations with and without rSC were not statistically different (see Results), we were able to treat the TS neurons as independent encoders of position and therefore pooled them across datasets. This, however, was not the case for the ELL populations which are redundant when correlations are taken into account (Haggard and Chacron, 2023). We, therefore, pooled the responsive TS neurons across recordings then randomly sampled populations of 31 neurons and resampled these populations as described above. This allowed us to compare the TS and ELL at higher population sizes up to the number of simultaneously recorded ELL neurons.
Effects of nonresponsive neurons on FI
Because the spatial receptive fields of nonresponsive neurons were not significantly spatially dependent, we set their slopes to zero for all spatial locations when computing FI. Thus, for an independent neural population, nonresponsive neurons cannot add to the FI because their individual contribution, which is given by the slope squared divided by the spike-count variance, is then zero. However, if there are correlations (i.e., the population is not independent), then nonresponsive neurons can influence FI through the inverse covariance matrix. We considered two cases: (1) responsive neurons only and (2) responsive and nonresponsive neurons. In the first case, we have as follows:
As mentioned above, nonresponsive neurons cannot contribute to the FI when responses are independent (i.e., when all off-diagonal elements of the covariance matrix are zero). Thus, if there were no correlations, then the FI for all neurons (Case 2) would then be equal to the FI obtained when only considering responsive neurons (Case 1). Thus, it is sufficient to consider the difference between
It is also useful to separate the sum into “diagonal” and “off-diagonal” terms as follows:
Similarly, when only considering responsive neurons, we have the following:
A difference between
Mathematical model
Rationale and model structure
To further investigate how different features of the TS neural populations under study impact the information conveyed during prey localization, we developed a mathematical model. We used the same model framework as we did in a previous publication investigating ELL population coding (Haggard and Chacron, 2023). Specifically, we generated spatial receptive field functions as well as the covariance matrix that are necessary to compute the FI. This was done initially by ensuring that these closely resembled those seen experimentally as described below. We then systematically varied model parameters such as spatial receptive field properties as well as the structure of the covariance matrix to gain understanding as to how they impact information transmission at the population level.
Model spatial receptive fields
We first parametrized the experimentally obtained spatial receptive fields. Specifically, we subtracted the baseline firing rate from the spatial receptive field for each responsive neuron to remove the bias and then fitted a difference of Gaussians as follows:
For receptive field position μ, the data were fit with a normal distribution (mean, 0; SD, 1.04; R2 = 0.96; n = 9). For the widths, the receptive field surround width σs was always by definition greater than the receptive field center width σc. Therefore, we first considered the distribution of the receptive field center widths which was fitted with a gamma distribution (shape, 4.93; scale, 0.22; R2 = 0.98; n = 9). We then considered the distribution of the difference between the surround and center receptive field widths (i.e., σs − σc) which was fitted with a gamma distribution (shape, 0.49; scale, 2.41; R2 = 0.4; n = 8). We used this approach because we found no significant correlation between the receptive field center width and the difference between the surround and center receptive field widths (R = −0.14; p = 0.26; n = 72). Finally, for the amplitudes, we considered the magnitude (i.e., absolute values of αc and αs). The center amplitude magnitude distribution was fitted with a t location-scale distribution (location, 3.46; scale, 1.86; shape, 0.49; R2 = 0.92; n = 39). Overall, it was found that the surround amplitude magnitude was always smaller than the center amplitude magnitude. We thus considered the distribution of the difference between center amplitude magnitude and surround amplitude magnitude, which was fitted with a gamma distribution (shape, 1.65; scale, 1.40; R2 = 0.83; n = 8).
Selection of parameter values for model neuronal receptive fields
Selection of parameter values for the spatial receptive field for each model neuron was done as follows. For responsive neurons, the receptive field position, center width, and center amplitude magnitude were drawn randomly from the distributions describe above. The difference between surround and center widths was then drawn from the distribution described above. The surround width was obtained by adding the difference between surround and center width to the center width. The difference between the center amplitude magnitude and the surround amplitude magnitude was then drawn from the distribution above. The surround amplitude was then obtained by subtracting this difference from the center amplitude, and it was verified that the surround amplitude was smaller than the center amplitude. If this was not the case, then a new difference value was drawn, and the process was repeated until the surround amplitude was smaller than the center amplitude. For OFF-type cells, we used the same procedure but multiplied both center and surround amplitude values by −1. Approximately 30% of the responsive model neurons were randomly selected to be OFF-type, as seen experimentally. Finally, the baseline firing rate for each model neuron was drawn from the experimentally observed distribution and added to the spatial receptive field, and then any negative values were set to zero. Such rectification mimics that seen experimentally for TS neurons, particularly those with low baseline firing rates.
For nonresponsive neurons, the baseline firing rate was randomly drawn from the distribution described above, and the spatial receptive field was set to be equal to the baseline firing rate for all spatial locations. As seen experimentally, 65.6% of neurons were set to be nonresponsive.
Ordering of model neurons
The model neurons were ordered by ascending receptive field center position, with the nonresponsive neurons intermingled with the responsive neurons. A vector of channel numbers, also ordered and drawn from the fitted distribution of channel numbers from the data (gamma distribution, shape, 2.19; scale, 35.62; R2 = 0.94; n = 7), was used to calculate a relative anatomical distance between the model neurons as done experimentally. This was done to ensure that pairs of responsive neurons with smaller relative anatomical distances displayed more receptive field overlap than pairs with larger relative anatomical distances, as seen experimentally.
Model neuron variance
We first investigated the relationship between the variance and firing rate. This was done by pooling and visualizing the firing rates as a function of variances for all neurons at all locations. We then binned and averaged the data in increments of 1.6 spikes/s and fit the binned data with the following saturating function using a linear least-square fit:
Model neuron correlations and covariance
The model spike-count correlations (rSC) were obtained from baseline correlations (rBL). To begin, the baseline correlations were visualized as a function of relative anatomical distance as described above and split into same-type (ON–ON and OFF–OFF) and opposite-type (ON–OFF) pairs because they were not symmetric around zero and therefore could not be pooled. The two groups were fit separately with exponential fits,
FI calculation and effect of correlations
We generated spatial receptive fields and covariance matrices for the model neurons as described above and calculated the spatially dependent FI and the FI as a function of the population size in the same manner as that used for the experimental data. The effect of correlations on FI was also assessed in the same manner as that used for the experimental data.
Comparing model and experimental data
As mentioned above, the goal of building a mathematical model was to gain understanding as to how different features of the TS neural populations (e.g., spatial receptive fields, correlations, etc.) impact information transmission. It is however important to ensure that the model can accurately reproduce experimental results in the first place. We quantitatively assessed the similarity between the FI values obtained from the model and experimentally using the variance accounted for, which is given by
Effects of spatial receptive field heterogeneity
We first used our model to gain understanding as to how spatial receptive field heterogeneity influences information transmission. Nonlinear rectification of spatial receptive fields did not affect the qualitative nature of our results. Such rectification was therefore omitted in the following.
We considered the effects of heterogeneity in position, width, and amplitude. First, we focused on the heterogeneity of the receptive field center positions while fixing the amplitude and width of the receptive fields at the mean values of their respective distributions (amplitude, 3.46; width, 1.09). Initially, all receptive fields in the population were centered at 0 cm, such that the receptive fields completely overlapped, and then the spread of the receptive field positions was gradually increased by increasing the SD of the normal distribution from which the receptive field positions were drawn (SDs, 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.04, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2, 3, 4 cm; the experimental data correspond to an SD value of 1.04).
Next, we held the amplitude constant (amplitude, 3.5 spikes/s), allowed the position to vary per the normal distribution from the data, and gradually increased the SD of the receptive field widths distribution. In our experimental data, the distribution of widths was best fit by a gamma distribution. However, because this distribution is not symmetric, we replaced it with a normal distribution so that the inclusion of narrower and wider receptive fields occurred in approximately equal amounts (SDs, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7 cm; the data correspond to a value of 0.35 cm).
Finally, we held the receptive field widths constant, allowed the position to vary per the normal distribution from the data, and varied the amplitude heterogeneity. Again, because the distribution of amplitudes in the data is not symmetric, we were not able to simply use the same distribution and increase the distribution spread. Instead, we used a normal distribution centered at the center of weight of the original distribution and gradually increased the SD (SDs, 0.1, 0.5, 1, 1.5, 1.87, 2, 2.5, 3, 3.5, 4, 4.5, 5 spikes/trial; the data correspond to a value of 1.87 spikes/trial). Note that as each trial is 0.125 s in duration, there are 8 trials/s. Thus, for example, 2 spikes/trial corresponds to 16 spikes/s. For all these simulations, the surround widths and amplitudes were generated as described above in the section on model spatial receptive fields.
In each of these simulations, we used the spatially averaged FI calculation described previously with populations of 30 neurons and ran 1,000 simulations for each change in heterogeneity and then visualized the FI versus parameter heterogeneity curves for the correlated and independent cases. We also quantified synergy by calculating the percentage difference between the correlated and the independent FI as follows: synergy = 100 * (FIcorr − FIind.) / FIind. We note that redundancy occurs for negative values.
Investigating how differences between ELL and TS in terms of spatial receptive fields and correlations impact information transmission
We investigated which changes in parameters were necessary to transition from the redundant situation seen in the ELL (Haggard and Chacron, 2023), where the spike-count correlations significantly reduce the information in the population activity relative to an independent population, to the synergy that we found in TS in the current paper (see Results). We, therefore, took a simplified ELL-like model, using the parameters and functions from a previously published model (Haggard and Chacron, 2023). We simplified the model by using the variance function fit to the TS data and keeping the proportion of ON- to OFF-type neurons from TS. We then changed one parameter distribution or function at a time to the TS parameter distribution or function and ran the full analysis. We also tried different combinations until we found which parameters are necessary to change from an ELL-like to a TS-like structure as described in the results.
Statistics
Unless otherwise indicated, values were reported as mean ± SEM, and statistical tests were performed using parametric or nonparametric tests as indicated in the text, depending on the normalcy of the data per the Lilliefors test. Where indicated, extreme outliers were removed using the generalized extreme studentized deviate test for outliers in MATLAB.
Data and code accessibility
All data files and codes are available from the Borealis repository (https://doi.org/10.5683/SP3/OUFXKH).
Results
The goal of this study was to understand how midbrain electrosensory neural populations within TS encode object location. To do so, a 4 Hz sinusoidal amplitude modulation of the animal's own EOD was delivered by a small dipole whose location was systematically moved along the animal's rostrocaudal axis while recording the activities of TS neural populations with a Neuropixels probe (Fig. 1). We recorded the activities of 209 TS neurons over 11 different sessions and quantified the accuracy of object localization by the FI which is obtained from the neural spatial tuning function (i.e., the spatial receptive field) and the covariance matrix of the neural responses (Abbott and Dayan, 1999). TS neurons receive input from ELL pyramidal cells (Fig. 1, bottom left, black arrows) and have extensive within-layer collateral connectivity as well as across-layer vertical connectivity. TS neurons project to higher brain areas mediating perception and behavior (Fig. 1, bottom left, black arrows).
Experimental setup. The fish is stimulated locally through a dipole electrode (top left) with a 4 Hz sinusoidal amplitude modulation (EOD black, AM red) while recording the activities of TS neurons simultaneously with a Neuropixels probe. The stimulus produces a spatially localized electric image which projects onto the surface of the fish's skin (white circles). The dipole delivering the stimulus is placed at various positions along the animal's rostrocaudal axis (arrow indicates direction as dipole is repositioned) both outside and within the receptive fields of the neurons being recorded (schematic receptive field on fish's skin in blue). Example spike trains of simultaneously recorded neural activity are shown on the right (dark blue). The early feedforward circuitry of the electrosensory system (bottom left) is shown with TS in the midbrain highlighted in yellow. EAs, electroreceptor afferents; ELL, electrosensory lateral line lobe; TS, torus semicircularis.
TS neural populations display heterogeneous receptive fields
We first mapped the spatial receptive fields of TS neurons as a function of stimulus position along the rostrocaudal axis. We only considered responses during the positive half-cycle of the stimulus, as the associated stimulus waveform mimics the increase in EOD amplitude caused by prey stimuli. We found that TS neurons could be categorized into three types: ON-type, OFF-type, and nonresponsive. Figure 2 shows the spiking activities of example neurons for each type in response to stimulation. For ON-type neurons, the strongest response consisted of increased spiking activity from the baseline firing rate (Fig. 2A, BL, thin black line). For OFF-type neurons, the strongest response consisted of decreased spiking activity from the baseline (Fig. 2B). When the dipole was moved to either side, we found that spiking activity had the opposite response relative to the baseline. As such, both ON- and OFF-type neurons displayed antagonistic center-surround organization, similar to what is found in ELL (Bastian et al., 2002). However, many TS neurons did not display changes in spiking activity in response to the stimuli used in this study (i.e., were nonresponsive; Fig. 2C). We found that most of the recorded neurons (65.6%) were nonresponsive. Furthermore, ON-type responsive neurons were more common than OFF-type neurons (23.9 and 10.5%, respectively). This significantly differs from results obtained previously in ELL where neurons were evenly partitioned between ON- and OFF-type and with most if not all neurons being responsive (Haggard and Chacron, 2023).
Mapping the spatial receptive fields of TS neurons. Neural responses from example TS neurons to stimulation at different locations on the rostrocaudal axis. Left panels, The AM (red) of the EOD is sinusoidal and the neural responses to the positive half-cycle were analyzed (the negative half-cycle is masked with the gray band). Below the stimulus waveform, raster plots of 20 example trials from two stimulus positions are shown with each black dot representing a single action potential. Below the raster plots, peristimulus time histograms (PSTH) show the responses across all trials. Right panels, Trial-averaged firing rates (blue) as a function of stimulus position (i.e., the receptive field) with shaded error bars indicating the SEM. The right and left black dots indicate the stimulus positions corresponding to the right and left raster plots and PSTHs, respectively. The baseline firing rate (BL, thin black line) is also shown to emphasize the antagonistic receptive field center and surround. A, Example ON-type neuron. This neuron displayed an increased firing rate from the baseline to stimulation within the receptive field center and decreased firing rate from the baseline to stimulation within the surround. Approximately 23.9% of neurons recorded in TS were ON-type. B, Example OFF-type neuron. This neuron displayed a response profile that was opposite to that of the neuron shown in A. Specifically, the firing rate decreased from the baseline in response to stimulation within the receptive field center and increased from the baseline in response to stimulation within the receptive field surround. Approximately 10.5% of recorded neurons were OFF-type. C, Example nonresponsive neuron. The firing rate for this neuron did not change significantly from the baseline during stimulation for all locations. Approximately 65.6% of neurons recorded in TS were nonresponsive.
TS neural populations display significant correlations during object-like stimulation
We next quantified covariances and correlations between TS neural activities. The covariance matrix from an example recording session at one stimulus location shows a wide range of values (Fig. 3A, top right, magenta triangle). To determine if these values were significantly different from zero, we recomputed them after randomly shuffling responses obtained during different stimulus trials (see Materials and Methods; Fig. 3A, bottom left, green triangle). Overall, covariance was distributed over a significantly greater range of values before than after randomly shuffling responses (Fig. 3A, compare top and bottom triangles; Kolmogorov–Smirnov test: p = 6.05 · 10−54; n = 1,225; test statistic = 0.32; Fig. 3B; Kolmogorov–Smirnov test: p = 4.82 · 10−83; n = 5.91 · 104; test statistic = 0.19). This was reflected in significantly higher covariance magnitude before random shuffling (Fig. 3B, inset; Kolmogorov–Smirnov test, p = 1.49 · 10−250; n = 5.91 · 104; test statistic = 0.34). Similar results were obtained when computing spike-count correlations (Fig. 3C; Kolmogorov–Smirnov test, p = 3.28 · 10−61; n = 1,225; test statistic = 0.34; Fig. 3D, Kolmogorov–Smirnov test, p =3.53 · 10−223; n = 5.91 · 104; test statistic = 0.34; Fig. 3D, inset: Kolmogorov–Smirnov test, p = 9.91 · 10−314; n = 5.91 · 104; test statistic = 0.65), which necessarily drive the significant covariances in the intact data because variances are not affected by shuffling. Further analysis revealed that correlations were not spatially dependent (spatial variation was only at most ∼4%) and decayed rapidly with distance between neurons (Fig. 4A). Importantly, the distribution of distances obtained when considering pairs of responsive neurons, pairs consisting of one responsive and one nonresponsive neuron, and pairs of nonresponsive neurons all strongly overlapped with one another (Fig. 4B). This suggests that nonresponsive and responsive neural populations were fully intermingled rather than being located at separate locations within TS.
Populations of TS neurons show significant covariance and spike-count correlation during stimulation. A, Example covariance matrix of a population of simultaneously recorded TS neurons at one stimulus location. The intact data (top right, magenta triangle) show significantly larger covariation than data obtained after randomly shuffling trials, which renders neural activities independent of one another (i.e., uncorrelated; bottom left, green triangle; Kolmogorov–Smirnov test, p = 6.05 · 10−54; n = 1,225; test statistic = 0.32). B, Probability density functions for covariance from our data before (magenta) and after (green) randomly shuffling trials (Kolmogorov–Smirnov test, p = 4.82 · 10−83; n = 5.91 · 104; test statistic = 0.19). Data were pooled over all simultaneously recorded pairs at all stimulus positions. Inset, The whisker-box plots of covariance magnitudes before (magenta) and after (green) randomly shuffling trials were significantly different from one another (Kolmogorov–Smirnov test, p = 1.49 · 10−250; n = 5.91 · 104; test statistic = 0.34; outliers not shown). C, Spike-count correlation (rSC) matrix from an example population of simultaneously recorded TS neurons at one stimulus location. The intact data (magenta triangle) display significantly larger correlations than after randomly shuffling the trials (green triangle; Kolmogorov–Smirnov test, p = 3.28 · 10−61; n = 1,225; test statistic = 0.34). D, Probability density function before (magenta) and after (green) randomly shuffling trials (Kolmogorov–Smirnov test, p = 3.53 · 10−223; n = 5.91 · 104; test statistic = 0.34). Data were pooled over all simultaneously recorded pairs at all stimulus positions. Inset, Whisker-box plots showing that correlation magnitude values obtained before (magenta) were significantly higher than values obtained after randomly shuffling trials (green; Kolmogorov–Smirnov test, p = 9.91 · 10−314; n = 5.91 · 104; test statistic = 0.65; outliers not shown). “*” indicates statistical significance at the p = 0.05 level.
Responsive and nonresponsive neural populations strongly overlap with one another in TS. A, Pairwise correlations in the absence of stimulation decay rapidly as a function of physical distance between neurons irrespective of whether the pair consists of responsive neurons (i.e., “resp. pairs,” blue), a responsive and a nonresponsive neuron (i.e., “resp.–non-resp. pairs,” green), or nonresponsive neurons (i.e., “non-resp. pairs,” light red). B, The distribution of pairwise distances obtained from responsive pairs (blue) strongly overlapped with those obtained from responsive–nonresponsive pairs (green, 83%) and from nonresponsive pairs (red, 73%). Distance distributions obtained from nonresponsive pairs and from responsive–nonresponsive pairs also strongly overlapped (87%).
TS neurons interact in a synergistic manner to increase information about stimulus location
We evaluated the accuracy of a linear decoder at determining stimulus location from the recorded spiking activities of TS neurons. Specifically, we computed the FI which is given by the inverse variance of the optimal estimator. FI is obtained from the slopes of the spatial receptive field and the covariance matrix. Values were averaged over the spatial region for which the recorded neurons were maximally informative to study the spatially averaged FI as a function of the population size and composition (Fig. 5A; see Materials and Methods). To quantify the effects of neural interactions on population coding, we compared FI values obtained from the intact data with spike-count correlations (w/rSC) expressed in the covariance matrices discussed above with those obtained from an independent population where the off-diagonal terms of the covariance matrix were set to zero (w/o rSC). In both cases, we found that the FI grows as a function of population size (Fig. 5B, blue and dark gray, respectively). Interestingly, however, FI values obtained from our dataset were significantly higher than those obtained from the independent population, indicating that interactions between TS neurons increase information via synergy (Fig. 5B, inset; Wilcoxon rank sum test, p = 6.37 · 10−11; n = 50; z statistic = 6.53, test statistic = 3,473).
Nonresponsive neurons increase information transmission by TS populations through synergistic interactions with responsive neurons. A, Schematic for calculating FI from neural receptive fields (ON-type, OFF-type, and nonresponsive) and their covariance matrices. B, Spatially averaged FI as a function of the population size with (w/rSC in blue) and without correlations (w/o rSC in dark gray). Inset, Whisker-box plots showing FI for a population of 36 neurons. FI was significantly higher with correlations than without (Wilcoxon rank sum test, p = 6.37 · 10−11; n = 50; z statistic = 6.53; test statistic = 3,473). C, Spatially averaged FI as a function of the population size but obtained when first adding only responsive neurons and then nonresponsive neurons. Inset, Whisker-box plots showing FI for responsive neurons only. FI values obtained before and after randomly shuffling trials were not significantly different from one another (Wilcoxon rank sum test, p = 0.08; n = 50; z statistic = −1.77; test statistic = 1,028). “*” indicates statistical significance at the p = 0.05 level; ns, not significant.
Responsive TS neurons independently provide information as to stimulus location, while synergy is caused by interactions between responsive and nonresponsive neurons
How do responsive and nonresponsive neurons contribute to synergistic coding of object location? This is an important question because, while nonresponsive neurons cannot contribute to the FI in the independent case due to the receptive field slope being zero, they can in theory contribute in the correlated case via the covariance matrix. Thus, we first considered populations of only responsive neurons and then added nonresponsive neurons (Fig. 5C). When only responsive neurons were considered, we found no significant differences in FI values from the intact and independent data (Fig. 5C, inset; Wilcoxon rank sum test, p = 0.08; n = 50; z statistic = −1.77; test statistic = 1,028). Therefore, the responsive neurons act as independent encoders, meaning that correlations, while present, do not impact the information they convey. The addition of nonresponsive neurons further increases information transmission through synergistic interactions (Fig. 5C).
Nonresponsive TS neurons increase information transmission by reducing the effective noise levels of responsive TS neurons
How do nonresponsive neurons increase information transmission? It is important to note that nonresponsive neurons are not spatially tuned to stimulus location and thus do not transmit information as to stimulus location because the slope of their spatial receptive field is zero. Thus, nonresponsive neurons can only influence information about stimulus location through interactions with responsive neurons. Specifically, including nonresponsive neurons can change the value of the diagonal elements of the inverse covariance matrix associated with responsive neurons, which can be seen as the reciprocals of the effective noise (see Materials and Methods). Alternatively, including nonresponsive neurons can change the value of the off-diagonal elements of the inverse covariance matrix. In this case, including nonresponsive neurons would affect the noise associated with interactions among responsive neurons. We note that these two possibilities are not mutually exclusive.
In this case, however, our results have shown that interactions between responsive neurons had negligible effect on FI (Fig. 5C). As such, we would expect that including nonresponsive neurons would affect the diagonal as opposed to the off-diagonal terms of the inverse covariance matrix. Confirming our prediction, the inverse covariance matrix obtained when including all neurons (i.e., responsive and nonresponsive; Fig. 6A, left) displayed diagonal elements that were lower than those of the inverse covariance matrix obtained when including responsive neurons only (Fig. 6A, right). This implies that the reciprocals of these elements (i.e., the “noise”; see Materials and Methods) is lower when nonresponsive neurons are included. Indeed, datapoints obtained when plotting the noise when only responsive neurons are included as a function of the noise when all neurons are included for each responsive neuron were consistently above the identity line (Fig. 6B), leading to a significant difference (Fig. 6B, inset; Wilcoxon signed rank test, p = 5.36 · 10−15; n = 81; z statistic = −7.82; test statistic = 0).
Nonresponsive neurons improve information transmission by effectively reducing noise for responsive neurons. A, Inverse covariance matrices shown for responsive neurons for the same location as shown in Figure 3. These were obtained when including responsive and nonresponsive neurons (i.e., “resp. + non-resp.,” left) or when only including responsive neurons (i.e., “resp.,” right). In both cases, the diagonal terms are outlined in green, and the off-diagonal terms in beige. Note that the diagonal terms are higher when all neurons are included (e.g., compare bottom-right diagonal terms for both matrices). B, Noise values, which are the reciprocal of the diagonal terms in the inverse covariance shown in A when considering responsive neurons only as a function of those obtained when considering all neurons for the diagonal terms (i.e., green in A). Datapoints were all above the identity line (dashed black). Inset, Whisker-box plot showing that differences between noise values obtained when considering responsive neurons only and those obtained when considering all neurons were significantly larger than zero (Wilcoxon signed rank test, p = 5.36 · 10−15; n = 81; z statistic = −7.82; test statistic = 0). C, Same as B, except that noise values were obtained as the reciprocals of the off-diagonal terms of the covariance matrices in A. Datapoints were distributed evenly around the identity line (dashed black). Inset, Whisker-box plot showing that differences between noise values obtained when considering responsive neurons only and those obtained when considering all neurons were not significantly larger than zero (Wilcoxon signed rank test, p = 0.88; n = 430; z statistic = −0.16; test statistic = 24,476). The data in B and C are pooled over the same locations as those used to calculate the spatially averaged FI in Figure 5. “*” indicates statistical significance at the p = 0.05 level; ns, not significant.
In contrast, noise values obtained as the reciprocals of the off-diagonal elements were similar in both conditions. Specifically, datapoints when plotting noise values obtained when considering responsive neurons only as a function of those obtained when considering all neurons were scattered around the identity line (Fig. 6C), leading to a difference that was not significantly different than zero (Fig. 6C, inset; Wilcoxon signed rank test, p = 0.88; n = 430; z statistic = −0.16; test statistic = 24,476). Therefore, our results demonstrate that nonresponsive neurons increase FI by effectively reducing the noise of responsive neurons, as opposed to affecting interactions between responsive neurons.
Populations of responsive TS neurons transmit more information about stimulus location than ELL pyramidal cells
How does population coding of object location change from ELL to TS? To address this question, we compared FI values obtained from responsive TS and ELL neurons. Overall, TS neurons were not individually more informative than ELL neurons as maximum FI values were not significantly different from one another (Fig. 7A, blue and purple, respectively; t test, p = 0.45; n = 158; df = 228; SD = 0.47; test statistic = −0.75). Next, we compared FI values as a function of population size and found significantly higher values for TS (Fig. 7B; Wilcoxon rank sum test, p = 7.01 · 10−18; n = 50; z statistic = 8.61; test statistic = 3,775). (Note that the TS curve is a lower bound on the FI because the synergy introduced by the nonresponsive neurons is not accounted for in this calculation.) The physiological implication of this result is that fewer responsive neurons should be required in TS to estimate the location of the prey with sufficient accuracy. To test this hypothesis, we converted the FI to the root Cramér–Rao bound which quantifies the SD of the optimal estimator. Overall, ∼13 and ∼19 neurons are sufficient to estimate the prey location in TS and ELL, respectively (Fig. 7C, blue and purple, with the average prey radius, 0.15 cm, marked with the light gray line).
TS neurons transmit more information as to object location than ELL neurons. A, Whisker-box plots showing FI values obtained for single responsive TS (blue) and ELL (purple) neurons, at the location where each neuron is maximally informative. No significant differences were observed (t test, p = 0.45; n = 158; df = 228; SD = 0.47; test statistic = −0.75). B, Spatially averaged FI as a function of population size for TS (blue) and ELL (purple). Inset: Whisker-box plots showing FI for a population size of 30 neurons for TS (blue) and ELL (purple). Significantly larger values were obtained for TS (Wilcoxon Rank Sum test, p = 7.01 · 10−18; n = 50; z statistic = 8.61; test statistic = 3,775). C, Root Cramér–Rao bound
Modeling TS population activities
So far, we have shown that responsive TS neurons act as independent encoders of stimulus location making them more informative than populations of ELL neurons. We also showed that nonresponsive TS neurons increase the information in the population through their contributions to the covariance matrices, thereby causing synergy. To better understand how heterogeneities in the population (i.e., receptive field parameters, correlation magnitudes and inclusion of nonresponsive neurons) impact the information contained in the neural activity, we next developed a mathematical model. We initially focused on making our model as similar to the experimental data as possible and then varied parameter values to ascertain their effect on population coding of object location. In a separate set of simulations, we used the model to better understand how differences between ELL and TS explain differential results obtained when considering population coding of object location. We used a model to explore these parameter spaces because it is not feasible to manipulate them experimentally.
Our model consisted of a population of model neurons with spatial receptive fields and covariances. Spatial receptive fields for responsive neurons were given by a difference of Gaussians (Fig. 8A) with receptive field position (cyan), width (center in dark green and surround in light green), and amplitude (center in dark orange and surround in light orange) drawn from distributions that were fitted to those obtained experimentally (Fig. 8B–H). The receptive field positions were normally distributed (Fig. 8B). The widths of the surround Gaussians were necessarily larger than the center Gaussians (Fig. 8C); therefore we fit the center distribution (Fig. 8D) and the distribution of differences between the surround and center (Fig. 8E). Similarly, the amplitudes of the center were always greater than that of the surround (Fig. 8F), so we fit the center amplitude distribution (Fig. 8G) and the distribution of the differences between the center and surround (Fig. 8H). Nonresponsive neurons had receptive fields with zero slope and firing rates equal to baseline firing rates drawn from a distribution fit to the data.
Parametrizing receptive field properties seen experimentally. A, Schematic of the sum of two Gaussians with the parameters corresponding to the data in B–H (position, cyan; center width, dark green; surround width, light green; center amplitude, dark orange; surround amplitude, light orange). B, Probability density function (pdf) obtained experimentally for receptive field position (cyan) and best-fit normal distribution (black; mean, 0; SD, 1.04; R2 = 0.96; n = 9). C, Surround width as a function of center width for our experimental data (green dots). It is seen that all datapoints lie above the identity line (dashed). D, Pdf of receptive field center width obtained experimentally (dark green) and best-fit gamma distribution (black; shape, 4.93; scale, 0.22; R2 = 0.98; n = 9). E, Pdf of the difference between receptive field surround and center widths obtained experimentally (light green) and best-fit gamma distribution (black; shape, 0.49; scale, 2.41; R2 = 0.4; n = 8). F, Surround amplitude magnitude as a function of center amplitude magnitude for our experimental data (orange dots). It is seen that all datapoints lie below the identity line (dashed). G, Pdf of center amplitude magnitude obtained experimentally (dark orange) and best-fit t location-scale distribution (black; location, 3.46; scale, 1.86; shape, 0.49; R2 = 0.92; n = 39). H, Pdf of the difference between receptive field center amplitude magnitude and receptive field surround amplitude magnitude obtained experimentally (light orange) and best-fit gamma distribution (black; shape, 1.65; scale, 1.40; R2 = 0.83; n = 8).
We then fit the variance as a saturating function of the firing rate (Fig. 9A), the spike-count correlations as a linear function of the baseline correlations (Fig. 9B), and the baseline correlations as an exponential function of the relative anatomical distance (Fig. 9C, left panel, same-type pairs; right panel, opposite-type pairs; see Materials and Methods). Together the variance and correlations were used to generate covariance matrices similar to the data.
Parametrizing key dependencies of variance and correlations seen experimentally. A, The variance was modeled as a saturating function (red curve) of the mean firing rate after binning and averaging the data (black dots). B, The spike-count correlations were not spatially modulated but did show a linear relationship with the baseline correlations (black dots), so they were modeled as a linear function of rBL (red linear fit) to which noise was added. C, However, to have a functional relationship between the spike-count correlations and the baseline correlations, we also had to model the baseline correlations as a function of distance between neurons in a given pair. To do so, we fit exponential functions (red curves) to first the same-type pairs (left panel) and the opposite-type pairs (right panel) after binning and averaging the data (black dots).
Before exploring how heterogeneity and correlations impact population coding, we compared results obtained from the model with those obtained experimentally (compare Fig. 10 with Fig. 5) and found excellent agreement as quantified by high variance accounted for values (94.22 ± 0.91%). Indeed, model FI curves increased as a function of the population size and were higher than those obtained for independent model populations (Fig. 10B; Wilcoxon rank sum test, p = 1.89 · 10−14; n = 50; z statistic = 7.66; test statistic = 3,634). Importantly, synergy was due to interactions between responsive and nonresponsive neurons, with the former acting as independent encoders (Fig. 10C and inset; Wilcoxon rank sum test, p = 0.72; n = 50; z statistic = −0.36; test statistic = 2,472).
Modeling population coding of object location by TS. A, Schematic of the model for which neural receptive fields (left colored curves, including ON-type, OFF-type, and nonresponsive neurons) and covariance matrices are used to calculate the FI. B, Spatially averaged FI as a function of the population size obtained from the model population with correlations (cyan) and the independent model population (light gray). Inset, Whisker-box plots showing FI for a population of 36 model neurons. FI was significantly higher for the model population with correlations than for the independent model population (Wilcoxon rank sum test, p = 1.89 · 10−14; n = 50; z statistic = 7.66; test statistic = 3,634). C, FI as a function of the population size but obtained when first adding only responsive model neurons and then nonresponsive model neurons. Inset, Whisker-box plots showing FI for responsive model neurons only. FI values obtained with and without correlations were not significantly different from one another (Wilcoxon rank sum test, p = 0.72; n = 50; z statistic = −0.36; test statistic = 2,472). “*” indicates statistical significance at the p = 0.05 level; ns, not significant.
Increased receptive field heterogeneity improves synergistic coding of object location
We next used our model to understand how receptive field heterogeneity influences the effects of correlations on population coding. The three parameters that are used to model the receptive fields are position, width, and amplitude. We started with studying position heterogeneity by fixing the width and amplitude and gradually increasing the heterogeneity of the position (Fig. 11A, left panel) from very low heterogeneity (left column, with all receptive fields centered around the same position) to high levels of heterogeneity (right column, with the receptive fields spatially tuned to different positions). We show that the FI as a function of position heterogeneity (Fig. 11A, middle panel) increases to a maximum and then gradually decays as the receptive fields become too spatially distant to have an additive affect. Interestingly, the data (gray vertical line) falls near the maximum for the correlated curve (cyan). It is also interesting to note that the correlated and independent curves (cyan and gray respectively) intersected causing a switch from a redundant regime to a synergistic regime near the heterogeneity level of the data (Fig. 11A, right panel). Next, we looked into heterogeneity in receptive field width (Fig. 11B) while keeping the amplitude fixed and using the position heterogeneity of the data. As the neural population shifts from receptive fields with very similar widths to more heterogeneous widths (Fig. 11B, left panel), the FI increases, indicating that a more heterogeneous population improves information transmission (Fig. 11B, middle panel). The FI curves with and without correlations also cross with increased width heterogeneity, but in this case the data fall at the point where redundancy is completely abolished. Finally, we considered the model parameter for receptive field amplitude (Fig. 11C). Increasing the heterogeneity of amplitude from low to high (Fig. 11C, left panel) increases the FI but only minimally and has little impact on the difference between the correlated and independent populations, with redundancy across all levels of heterogeneity (Fig. 11C, middle and right panels). Thus, our model shows that increased receptive field heterogeneity leads to synergy.
Increasing receptive field heterogeneity increases synergy. A population of 30 responsive TS neurons was modeled while varying the level of heterogeneity for three model parameters. Left panels, Schematic of parameter heterogeneity showing three example receptive fields with lower levels of heterogeneity on the left and higher heterogeneity on the right for each parameter. Middle panels, FI as a function of parameter heterogeneity with the correlated data in cyan and the independent data in light gray. The vertical line indicates the level of heterogeneity found in the data. Right panels, Synergy as a function of parameter heterogeneity. A, Receptive field position heterogeneity: the FI increases to a maximum then decays with increasingly higher levels of heterogeneity for both correlated and independent curves with the data falling near the maximum. Redundancy can be seen at lower levels of heterogeneity, but it then switches to synergy as the heterogeneity continues to increase. B, Receptive field width heterogeneity: the information increases monotonically with increased heterogeneity. Again, the relationship between the correlated and independent data switches from redundant to synergistic, and in this case the data fall at the approximate heterogeneity level where redundancy is abolished. C, Receptive field amplitude heterogeneity: increased heterogeneity gradually increases the FI, but the difference between the correlated and independent data is redundant and mostly independent of changes in amplitude heterogeneity.
A combination of increased receptive field heterogeneity, correlation structure, and nonresponsive neurons are sufficient to give rise to synergistic coding of object location
Lastly, we used the model to determine what parameters needed to be changed from an ELL-like model to give rise to the synergistic coding of object location seen in TS. We started with an ELL-like model [adapted from Haggard and Chacron (2023); see Materials and Methods], with the parameters and functions fit to ELL data, then systematically switched parts of the model to the parameters or functions fit to the TS data. The ELL-like FI increases as a function of the population size (Fig. 12A), but the independent data increase more rapidly than the correlated data due to the information limiting correlations in ELL, thereby causing redundancy (cyan and light gray, respectively). The first two changes we made to move from the ELL-like model toward the TS model were to individually switch the baseline correlation function and the position heterogeneity distribution (Fig. 12B,C, respectively). Separately, these changes decreased the distance between the correlated and independent FI curves; however, the effect was not strong enough to fully abolish the redundancy until we combined the two (Fig. 12D). Therefore, decreasing correlation magnitude and increasing position heterogeneity is sufficient to switch from ELL-like population coding to the population coding of the responsive TS neurons. Finally, we added nonresponsive neurons (Fig. 12E). (Note that the FI decreases markedly in Fig. 12E, but this is simply because we replaced responsive neurons with nonresponsive neurons to keep the population size the same across panels.) Together, these three modifications fully transitioned the model from a redundant regime to a synergistic one (Fig. 12F; one-sample t test p values, ELL-like 7.36 · 10−35; n = 50; df = 49; SD = 5.94; test statistic = −32.57; rBL = 8.80 · 10−30; n = 50; df = 49; SD = 6.57; test statistic = −25.30; pos. = 1.90 · 10−29; n = 50; df = 49; SD = 5.53; test statistic = −24.87; rBL + pos. = 0.44; n = 50; df = 49; SD = 6.23; test statistic = −0.78; rBL + pos. + n.r. = 3.78 · 10−17; n = 50; df = 49; SD = 7.53; test statistic = 12.73; TS model = 3.00 · 10−31; n = 100; df = 99; SD = 9.73; test statistic = 17.06). Taken together, changes in the correlation structure, increased receptive field heterogeneity, and the addition of nonresponsive neurons to the population are all necessary to give rise to synergistic population coding of object location by TS neurons.
Changes in correlation structure, together with increased RF heterogeneity and nonresponsive neurons are necessary to give rise to synergistic population coding of object location by TS neurons. A, An ELL-like model of FI as a function of the population size for correlated populations (cyan) and independent populations (light gray). Note that in this case, the FI is lower before as opposed to after randomly shuffling trials, indicating that correlations give rise to redundancy. B–E, Like A but gradually replacing one or more parameter or function from the ELL-like model with the equivalent from the TS model as follows: (B) baseline correlations (rBL), (C) position heterogeneity distribution, (D) rBL and position heterogeneity distribution, (E) rBL, position heterogeneity distribution and adding nonresponsive neurons to the population. F, The percentage difference, or synergy, for populations of 36 neurons for the five cases in A–E as well as the full TS model (one-sample t test, ELL-like p = 7.36 · 10−35; n = 50; df = 49; SD = 5.94; test statistic = −32.57; rBL = 8.80 · 10−30; n = 50; df = 49; SD = 6.57; test statistic = −25.30; pos. = 1.90 · 10−29; n = 50; df = 49; SD = 5.53; test statistic = −24.87; rBL + pos. = 0.44; n = 50; df = 49; SD = 6.23; test statistic = −0.78; rBL + pos.+n.r. = 3.78 · 10−17; n = 50; df = 49; SD = 7.53; test statistic = 12.73; TS model p = 3.00 · 10−31; n = 100; df = 99; SD = 9.73; test statistic = 17.06). “*” indicates statistical significance from zero at the p = 0.05 level; ns, not significant.
Discussion
Summary of results
We investigated how midbrain electrosensory neural populations encode prey location in weakly electric fish. Specifically, we recorded the simultaneous activities of multiple TS neurons in response to a local prey–mimic stimulus at different positions along the animal's rostrocaudal axis. We quantified the performance of an optimal unbiased estimator using the FI, which was computed from the neural spatial tuning functions (i.e., receptive fields) and covariances in spiking activities. Overall, we found that responsive TS neurons independently encoded object location, in spite of significant correlations. Interestingly, adding the activities of nonresponsive neurons, which cannot by themselves transmit information about object location, increased FI, thereby leading to synergistic coding of object location. Further analysis revealed that this was because the addition of nonresponsive neurons reduced the noise for individual responsive neurons rather than by modifying interactions between responsive neurons. Mathematical modeling showed that independent encoding by responsive neurons was due to greater tuning heterogeneity as well as reduced covariance magnitude.
TS neural populations use different strategies to encode object location than their afferent ELL neurons
Our results show that responsive TS neurons act as independent encoders of object location, while nonresponsive neurons improve information transmission via synergy. This contrasts with results obtained in afferent ELL populations, where responsive neurons instead displayed redundant coding of object location. One important question is then: Why do responsive TS neural populations encode object location independently while ELL neural populations instead display redundancy? Our mathematical model shows that this difference can be explained by the fact that TS neurons display more heterogeneous spatial tuning functions than ELL pyramidal cells. This result is consistent with modeling predications that demonstrate how positive correlations might negatively impact information transmission (i.e., introduce redundancy) in homogenous neural populations but instead positively impact information transmission (i.e., introduce synergy) in heterogeneous neural populations (Shamir and Sompolinsky, 2006; Ecker et al., 2011), as recently demonstrated in the primate vestibular system (Mohammadi et al., 2024).
Another important question is then: What is the contribution of nonresponsive ELL neurons to population coding of object location? To answer this, it is important to highlight differences between both areas. ELL pyramidal cells are somatotopically organized into columns with neurons within one column showing almost 100% receptive field overlap (Maler, 2009b). As such, objects at a given location on the fish's sensory epithelium will only elicit responses from ELL pyramidal cells whose receptive fields comprise this location (Shumway, 1989; Bastian et al., 2002; Maler, 2009a). However, the nonresponsive ELL pyramidal cells will be located some distance away from the responsive cells. Because correlations between ELL pyramidal cells decay as a function of distance (Chacron and Bastian, 2008; Hofmann and Chacron, 2017; Metzen and Chacron, 2021; Haggard and Chacron, 2023), it is likely that correlation magnitude between responsive and nonresponsive ELL neurons will be small. Therefore, adding nonresponsive ELL neurons will have, at best, minimal effect on population coding of object location. In contrast, TS neurons are organized in a somatotopic fashion but in multiple layers (Carr et al., 1981; Carr and Maler, 1985, 1986) and are also more selective in their responses than ELL neurons (Vonderschen and Chacron, 2011). As such, only a subset of TS neurons was responsive to object location, with the nonresponsive neurons sensitive to other stimulus categories (e.g., social interactions; Vonderschen and Chacron, 2011; McGillivray et al., 2012). Importantly, however, responsive and nonresponsive TS neurons largely overlapped with one another on the Neuropixels probe, implying that they could be located quite close to one another (Fig. 4). These correlations can then be high in magnitude and thus impact population coding, as observed in the current study.
It is important to note that our results showing that nonresponsive TS neurons can contribute to synergistic coding of object location differ from previous results obtained using electrocommunication signals. Indeed, a previous study showed that including nonresponsive neurons did not affect information transmission (Metzen and Chacron, 2021). We hypothesize that this is due to differences between the stimuli: the stimuli used in the current study of population coding of object location were spatially localized, whereas the electrocommunication stimuli used previously were spatially diffuse. Studies have shown that single ELL neurons respond differentially to spatially localized and diffuse stimuli (Bastian et al., 2002; Doiron et al., 2003; Chacron et al., 2005; Chacron, 2006; Marsat et al., 2009; Avila Akerberg and Chacron, 2010; Mejias et al., 2013). Moreover, ELL pyramidal cell populations display differential correlations to both stimulus categories (Chacron and Bastian, 2008; Litwin-Kumar et al., 2012; Simmonds and Chacron, 2015; Wang and Chacron, 2021). We hypothesize that whether nonresponsive neurons contribute or not to population coding will strongly depend on both single neuron response properties and correlations at the population level. Further modeling and experimental studies are needed to test this hypothesis.
We also found differences in correlations between TS and ELL neural populations. Correlations in ELL were high in magnitude which decayed with increased distance and were spatially dependent (Haggard and Chacron, 2023). In contrast, TS displayed lower magnitude correlations that also decayed with increased distance but were spatially independent. We hypothesize that these differences in correlations are the result of differences in the anatomical organization of ELL and TS (Rosenbaum et al., 2017). While ELL pyramidal cell correlations are most likely due to shared input from peripheral EA afferents (Chacron and Bastian, 2008), it is likely that TS correlations are influenced by extensive lateral (i.e., within layer) and vertical (i.e., across layer) connectivity (Carr et al., 1981; Carr and Maler, 1985). Further studies are needed to understand how differences in anatomical organization contribute to the observed differences in correlation between TS and ELL and are beyond the scope of this study.
Is synergistic coding of object location by TS neurons decoded downstream?
Ultimately, the type and amount of information that is present in the activity of a neural population are only of relevance to the organism if it is decoded by downstream brain areas (Panzeri et al., 2022). Therefore, the synergistic contribution of nonresponsive TS neurons can only improve information transmission if downstream neurons receive converging input from responsive and nonresponsive TS neurons. A recent study suggests that this is indeed the case by demonstrating that neurons in the PG, which receive extensive direct input from TS, display mixed selectivity (Wallach et al., 2022). Mixed selectivity has been observed across systems and species [e.g., memory and cognition in monkeys (Rigotti et al., 2013); visual and motor processes in monkeys (Maranesi et al., 2024); navigation in mice (Kira et al., 2023)] and is hypothesized to provide a high-dimensional space for reliably encoding complex sensorimotor features (Fusi et al., 2016; Johnston et al., 2020). Mixed selectivity occurs when neurons respond to a combination of stimulus features from different behavioral contexts. In the electrosensory system, PG neurons respond to a combination of stimulus features from object location, electrocommunication, and other conspecific interactions. Perhaps the simplest explanation for such mixed selectivity is that PG neurons receive input from TS neurons that respond selectively to stimuli corresponding to these behavioral contexts. It is thus physiologically plausible that PG neurons would receive input from TS neurons that are responsive and nonresponsive to object location, which would allow for increased information as to object location and potentially contribute to the animal's extraordinary abilities at prey detection (Nelson and Maciver, 1999). Further studies are however needed to test these predictions.
Nonresponsive neurons can contribute to population coding
Neurons that are nonresponsive to the stimuli used in any given study are frequently dismissed; they may be excluded through selection bias (i.e., during data collection or preprocessing) or reported but excluded from analyses as it is assumed that they do not contribute (Pasupathy and Connor, 2002; Fiscella et al., 2015; Franke et al., 2016; Nigam et al., 2019; Ni et al., 2022). While it is true that nonresponsive neurons cannot contribute to FI through their spatial tuning functions, our results show that these can indeed contribute by effectively reducing noise for responsive neurons. This is, to our knowledge, the first experimental demonstration of such a phenomenon. Previous studies have highlighted important similarities between the electrosensory and other sensory systems [e.g., vision; see Clarke et al. (2015) for review]. Similarly, the presence of nonresponsive neurons intermixed with responsive neurons is not unique to TS. Studies have investigated the affect of neuromodulators on the strength of neural selectivity for echolocation stimuli as opposed to tones or noise stimuli in the inferior colliculus (i.e., the auditory equivalent to TS) of bats (Fuzessery and Hall, 1996; Hurley and Pollak, 1999; Fuzessery et al., 2006). In the olfactory sensory system, the piriform cortex contains neurons which respond selectively to a single or a select combination of odorants in rats (Yoshida and Mori, 2007; Zhan and Luo, 2010). Additionally, in the visual system, the inferior temporal cortex in monkeys contains a population of neurons with high selectivity for specific visual inputs that do not respond in the absence of their preferred stimuli, intermingled with invariant neurons (Rust and Dicarlo, 2010, 2012). It is likely that our results will be applicable to other systems.
Footnotes
This work was supported by Canadian Institutes of Health Research (Grant Number MOP-159694) and Natural Sciences and Engineering Research Council (RGPIN-2020-04199), received by M.J.C.
The authors declare no competing financial interests.
- Correspondence should be addressed to Maurice J. Chacron at maurice.chacron{at}mcgill.ca.