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The Journal of Neuroscience, October 15, 2002, 22(20):9053-9062

Bursting Neurons Signal Input Slope

Adam Kepecs, Xiao-Jing Wang, and John Lisman

Volen Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02454

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Brief bursts of high-frequency action potentials represent a common firing mode of pyramidal neurons, and there are indications that they represent a special neural code. It is therefore of interest to determine whether there are particular spatial and temporal features of neuronal inputs that trigger bursts. Recent work on pyramidal cells indicates that bursts can be initiated by a specific spatial arrangement of inputs in which there is coincident proximal and distal dendritic excitation (Larkum et al., 1999). Here we have used a computational model of an important class of bursting neurons to investigate whether there are special temporal features of inputs that trigger bursts. We find that when a model pyramidal neuron receives sinusoidally or randomly varying inputs, bursts occur preferentially on the positive slope of the input signal. We further find that the number of spikes per burst can signal the magnitude of the slope in a graded manner. We show how these computations can be understood in terms of the biophysical mechanism of burst generation. There are several examples in the literature suggesting that bursts indeed occur preferentially on positive slopes (Guido et al., 1992; Gabbiani et al., 1996). Our results suggest that this selectivity could be a simple consequence of the biophysics of burst generation. Our observations also raise the possibility that neurons use a burst duration code useful for rapid information transmission. This possibility could be further examined experimentally by looking for correlations between burst duration and stimulus variables.

Key words: burst; biophysical model; pyramidal cell; weakly electric fish; ELL; neural coding; simulation

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

A fundamental aspect of neuronal function is how the output spike pattern of a neuron is determined by its synaptic input. A simple and highly successful formulation of this process has been the "integrate-and-fire" model (Lapicque, 1907). According to this model, neurons integrate synaptic input via their membrane capacitance and fire spikes when their voltage reaches spike threshold (Lapicque, 1907; Tuckwell, 1988). For the many types of neurons that obey this principle, the rate of spiking directly reflects the amplitude of the input current (Kistler et al., 1997; Binder et al., 1999).

However, many neurons possess additional voltage-gated conductances that can participate in the generation of more complex firing patterns. For instance, intrinsic conductances can generate brief, high-frequency bursts of action potentials that are commonly observed in recordings from a variety of brain regions (Kandel and Spencer, 1961; Barker and Gainer, 1975; King et al., 1976; Cattaneo et al., 1981a; Eisen and Marder, 1982; Gariano and Groves, 1988; Reinagel et al., 1999; Martinez-Conde et al., 2000; Ramcharan et al., 2000). There is evidence that such bursts represent a special neural code (Gabbiani et al., 1996; Lisman, 1997; Sherman, 2001). For instance, hippocampal place-fields are more accurately defined (Otto et al., 1991; Molden et al., 2001) (but see Harris et al., 2001), feature selectivity of some neurons in the visual cortex is sharpened (Cattaneo et al., 1981b; Livingstone et al., 1996), and feature extraction by electric fish pyramidal cells is more reliable (Gabbiani et al., 1996) when only spikes belonging to bursts are considered. In monkeys performing a motion discrimination task, the burst rate was found to reflect the direction of visual stimulus better than the average firing rate (Bair et al., 1994). In vitro, bursts of pyramidal neurons were found to underlie population synchrony in the cortex (Silva et al., 1991) and the hippocampus (Miles et al., 1988). Bursting has been shown to play crucial roles in synaptic plasticity both presynaptically (Pavlides et al., 1988; Huerta and Lisman, 1995) as well as postsynaptically (Thomas et al., 1998; Pike et al., 1999). Furthermore, some forms of short-term plasticity allow synapses to reliably transmit bursts but filter out single spikes (Lisman, 1997; Matveev and Wang, 2000). Given this evidence for the importance of bursts, it is crucial to understand which properties of neuronal input trigger burst firing.

We have examined this issue using a computational model of a common class of bursting neurons. We stimulated the model with sinusoidal and random current input and looked for the temporal input features that trigger bursts. Our main conclusion is that the membrane conductances that generate bursts can make the neuronal output dependent more on the slope of the input current than its amplitude. This result may explain previous experimental observations that some bursting neurons are involved in the behaviorally important process of detecting the slope of sensory signals (Gabbiani et al., 1996; Metzner et al., 1998; Sherman, 2001). Our work also revealed a novel aspect of burst-mediated signaling: the magnitude of the slope can be encoded by the number of spikes per burst. Because burst duration could transmit graded information in <20 msec, it may be an especially useful coding mechanism when rapid information transmission is at a premium.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Neuron model. We used a model neuron that includes only the minimal biophysical mechanisms necessary to reproduce bursting in a pyramidal cell (Traub et al., 1991; Pinsky and Rinzel, 1994; Kamondi et al., 1998; Mainen and Sejnowski, 1996; Doiron et al., 2001). We represent the soma and the axon lumped into one compartment containing the channels necessary for spike generation (I_Na and I_K), which result in a type I membrane (Hodgkin, 1948; Rinzel and Ermentrout, 1989; Wang and Rinzel, 1995; Wang and Buzsáki, 1996). The dendritic compartment includes a slowly activating potassium, I_KS, and a persistent sodium, I_NaP, current (Azouz et al., 1996), which together are responsible for burst generation. The membrane potential obeys the following equations (Kamondi et al., 1998): C_mdV_s/dt = I_Na  I_K  I_leak  g_c (V_s  V_d)/p + I_soma and C_mdV_d/dt = I_NaP  I_KS  I_Leak  g_c (V_d  V_s)/(1  p) + I_dendrite. The voltage-dependent conductances are described using standard Hodgkin-Huxley formalism. The kinetics of a gating variable x are described by dx/dt = _x(_x(1  x)  _xx) = _x(x  x)/_x. I_Na = g_Nam³h(V  E_Na), where m = _m/(_m + _m), _m = 0.1(V + 31)/(exp(0.1(V + 31))  1), _m = 4exp((V + 56)/18); _h = 0.07 exp((V + 47)/20), and _h = 1/exp(0.1(V + 17)) + 1). I_K = g_Kn⁴(V  E_K), where _n = 0.01(V + 34)/(exp(0.1(V + 34))  1), and _n = 0.125 ((V + 44)/80). I_NaP = g_NaPT³(V  E_Na), r(V) = 1/(1 + ((V + 57.7)/7.7)). I_KS = g_KSq(V  E_K), where q(V) = 1/(1 + exp((V + 35)/6.5)) and _q(V) = q0/(exp((V + 55)/30) + exp((V + 55)/30)). I_Leak = g_Leak(V  E_Leak), C_m = 1 µF/cm². The coupling conductance between compartments is g_c = 1 mS/cm². In Figures 2a, 3b, and 6, a and b, we only use the somatic compartment, g_c = 0, whereas in Figure 2d we obtain adaptation by setting g_c = 0.1. The asymmetry between the areas of the two compartments is taken into account in the parameter p = somatic area/total area = 0.15. The total membrane area is assumed to be 60,000 µm². The temperature scaling factors are _h = _n = 3.33. Other parameter values are: g_Leak = 0.18, g_Na = 45, g_K = 20, g_NaP = 0.12, and g_KS = 0.8 in mS/cm². In some simulations we modified the bursting conductances g_NaP = 0.09, g_KS = 0.9 to obtain intermixed bursts and spikes. The activation time constant of I_KS is scaled by _q0, which was set to 200 msec unless otherwise noted (see Fig. 3e). The ionic reversal potentials are E_Leak = 65, E_Na = +55, and E_K = 90 in mV. Numerical integration was performed with a fourth-order Runge-Kutta method using a 0.01 msec time step. Bifurcation analysis was done using AUTO (Doedel, 1981) in XPP (Ermentrout, 2002).

We simulated two other models that differ in the specific ionic mechanisms of burst generation but belong the same broad class of bursters. One model is based on a calcium-dependent bursting mechanism (Wong et al., 1979; Traub et al., 1991). In this model the dendritic compartment contains a high-threshold calcium current, I_CaL, and a calcium-dependent potassium current, I_AHP. The parameters are the same as in Wang (1998), except g_Ca = 0.35, g_AHP = 6.5, g_c = 1.5 (in mS/cm²), and p = 0.2. Another model tested is the chattering cell model as in Wang (1999), except g_NaP = 0.09 and g_KS = 10.0. Simulation code for these models can be found at http://www.bio.brandeis.edu/lismanlab/bursting/.

Stimulus generation. We generated random stimuli in the frequency domain with a flat power spectrum up to a given cutoff frequency, f_c. The cutoff frequency introduces correlations in the input at _corr = 1/2f_c. For instance, in Figure 4 the use of f_c = 5 Hz creates correlations at the time scale of 100 msec.

Spike train analysis. The spike trains in response to random current injections were segmented into bursts and single spikes based on an interspike interval (ISI) threshold. Bursts were defined as the set of spikes occurring within 10 msec of another spike. This ISI threshold was based on the ISI distribution, but small changes did not affect the qualitative results.

Signal detection theory. We characterized the differences between the coding properties of different bursts using the receiver-operating characteristic (ROC) (Green and Swets, 1966). We use ROC analysis to characterize how well certain aspects of the stimulus predict the burst output. This in turn allows us to evaluate how bursts of different durations signal different aspects of the stimulus. The ROC curve characterizes the range of decision options available for different detection criteria. For any set rate of false alarms, P_FA, the probability of correct detection, P_D, is plotted. The diagonal, P_FA = P_D, represents chance level detection. The more bowed a curve toward the top-left corner, the better the overall detectability.

The discriminability index is defined as the area under the ROC curve, signifying the probability of correct discrimination (Gabbiani and Koch, 1998). The discriminability between 2 and 3 spike bursts is defined as P_{D: 2-3} =  P(B2 = x) P(B3 < x) dx, where B2 and B3 denote two and three spikes bursts, respectively, and x is either the amplitude or the slope of the input. P(B2 = x) represents the probability that a two-spike burst occurs at x, whereas P(B3 < x) represents the probability that a burst of three spikes will occur below x. Here, x can represent any aspect of the stimulus, and we used either the amplitude or the slope of the current.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

To investigate the signaling properties of bursts we used a standard model of bursting pyramidal neurons (Fig. 1A) (Traub et al., 1991; Rhodes and Gray, 1994; Pinsky and Rinzel, 1994; Mainen and Sejnowski, 1996; Kamondi et al., 1998; Wang, 1999; Kepecs and Wang, 2000; Doiron et al., 2001). This model captures the general features of burst generation in pyramidal cells, in particular the interaction of the soma and the dendrites in burst generation (Wong et al., 1979; Turner et al., 1994; Hoffman et al., 1997; Golding et al., 1999; Williams and Stuart, 1999). The soma contains the classic Hodgkin-Huxley currents responsible for spike generation (a Na⁺ current with fast inactivation and a K⁺ current). The dendrite contains the currents responsible for bursting: a fast-activating, persistent inward current, I_NaP, and a slowly activating potassium current, I_KS (Fig. 1A). The role of the persistent inward current is to produce a regenerative current that drives the high firing rate (100-300 Hz) during a burst. Because the inward current is regenerative, a burst can outlast the brief inward current that triggers it (Fig. 1B). On the other hand, bursts terminate abruptly because of the slowly activating K⁺ current, even when the input is sustained (Fig. 1C). These two properties are characteristic of a broad class of bursting neurons. This model is just one implementation of the many possible ionic mechanisms for bursting that can be characterized by a fast positive feedback process (underlying Fig. 1B) coupled to slower negative feedback (underlying Fig. 1C).

View larger version (17K): [in this window] [in a new window]   Figure 1.   Model of a bursting neuron. A, Two-compartment model of pyramidal neurons. The somatic region contains Hodgkin-Huxley-type currents that generate fast spikes, and the dendrite contains a persistent sodium and a slow potassium current responsible for bursting. B, A brief current injection to the dendritic compartment induces a burst that outlasts the stimulus. This is a form of fast positive feedback. C, Burst firing terminates even for a sustained current injection. This is indicative of a negative feedback process. The current pulse is 6 msec in B and 225 msec in C.

Slope detection by bursts

Using this model we examined how the temporal properties of the neuronal input affect burst initiation. A traditional Hodgkin-Huxley neuron fires at a rate proportional to the input amplitude and thus fires symmetrically on both rising and falling phases of a 4 Hz sinusoidal input (Fig. 2A). In contrast, if the conductances that produce bursting are present, the cell fires bursts exclusively on the rising phase of the 4 Hz signal (Fig. 2B). These results gave the first indication that bursting neurons signal positive slope. Examination of the individual conductances provides insight into how selectivity to upstrokes is achieved: each spike of the burst produces a cumulative elevation of the slow dendritic potassium current, I_KS (Fig. 2C). This enhanced K⁺ current eventually terminates the burst and, because it turns off slowly, prevents firing on the falling phase of the input. It is these hysteretic properties that break the firing symmetry. Note that spike frequency adaptation produced by I_KS alone could break the symmetry (Fig. 2D). However, the bursting that is generated when a fast inward current is also present provides a very strong form of adaptation (Fig. 1C), enhancing slope detection. Without this inward current (fast positive feedback) the firing rate on the rising edge is low so there is little negative feedback and hence the difference between the rates on the rising and falling phases is not large. Additionally, the fast inward current produces the high spike rates that contribute to coding, as will be discussed later.

View larger version (36K): [in this window] [in a new window]   Figure 2.   Biophysics of bursts leads to detection of the up-stroke of 4 Hz current stimulus. A, Tonically spiking neuron. With only spiking currents (IHH) the firing in response to a sinusoid current is symmetric with respect to the peak of the input current. B, Bursting neuron. Bursts occur on the rising edge of the sinusoidal input. C, Mechanism of slope selectivity. After the burst is terminated, the slow IKS remains elevated and counteracts the input current, so that the net current flow is almost zero (dotted line represents 0 current). Calibration: 4 nA, 50 msec. D, Adapting neuron. Adaptation only weakly breaks the firing symmetry.

If bursting neurons are robust slope detectors, this capability should not be restricted to sinusoidal inputs. We therefore examined the behavior of the model with random input signals (Fig. 3A). Our simulations showed that spikes within bursts occurred preferentially on the positive slopes of stimuli, whereas isolated spikes showed little slope selectivity (Fig. 3B). We further found that the model neuron is capable of burst firing on consecutive up-strokes without an intervening down-stroke (Fig. 3A).

View larger version (43K): [in this window] [in a new window]   Figure 3.   Bursts are triggered on the positive slopes of randomly varying input. A, Random input (with cutoff at 5 Hz; mean 1.7; SD 1.0) was applied to the dendrite. Dendritic conductances were adjusted to gNaP = 0.09, gKS = 0.9 (mS/cm2) so the model fires single spikes as well (marked with dots). The top trace shows the random stimulus, and the bottom trace shows the membrane potential response. Bursts and single spikes were identified using a 10 msec ISI threshold, which was set based on the ISI distribution. Asterisks mark consecutive bursts without an intervening hyperpolarization. B, Histogram of input slopes at spike times. Burst spikes occur on positive slopes, whereas single spikes show little slope selectivity. For a tonically spiking model (soma only), the distribution of spikes over different slopes is symmetric for the same signal. C, Reverse correlations of the first spikes of bursts with random stimuli. The white noise stimuli had cutoff frequencies of 4, 10, and 30 Hz. Bursts are always preceded by an upstroke. D, Reverse correlations of bursts with the derivative of the stimulus. The cutoff frequency of the random stimulus was varied between 2 and 30 Hz. The reverse correlation for each frequency was normalized to the maximum slope (shown with dark red). E, Same reverse correlations as in D with a random stimulus having an 8 Hz cutoff. The factor, q0, which scales the activation time constant of the slow potassium conductance, is varied. The standard value is q0 = 200 msec.

Next, we used the reverse correlation technique to examine the features of the input that trigger bursts at different frequencies. We simulated the model neuron receiving a white noise input signal lowpass filtered at different cutoff frequencies. As shown in Figure 3C, bursts follow stimulus upstrokes over a wide range of frequencies. Figure 3D shows the mean slope of the stimuli preceding bursts at a range of frequencies. The maximal positive slopes (dark red) always precede bursts. Note, that at >10 Hz there is a frequency-dependent delay of bursts up to ~20 msec. This frequency-dependent delay could degrade the temporal precision of slope signaling. The frequency dependence of delay comes about because of the slow membrane properties and in particular because of the slow decay of the dendritic potassium current. There is no delay when the dominant input frequency approximately matches the activation time constant of I_KS (_q0 = 200 msec matches 5 Hz). Figure 3E shows how the burst delay can be tuned by varying the time constant of activation for I_KS. When the activation rate is slow compared with the stimulus frequency, bursts occur with a delay. Thus, the selectivity of bursts for specific stimulus features depends on the tuning of the burst conductances. Taken together, the results of Figures 2 and 3 demonstrate that the burst-mediated mechanism for slope detection is robust for a range of input statistics.

Bidirectional slope signaling

Our findings may be relevant to experimental studies on burst coding in weakly electric fish. Bursts in pyramidal cells of the electrosensory lateral-line lobe detect temporal changes in the surrounding electric field (Gabbiani et al., 1996; Metzner et al., 1998). This is accomplished by two different groups of cells: E-cells that signal the up-stroke and I-cells that signal the down-stroke of field modulations (Bastian, 1981; Shumway and Maler, 1989; Gabbiani et al., 1996). However, the mechanisms underlying this selectivity have been unclear. It is known that E-cells receive excitatory input directly from the electroreceptor afferents, so our previous results provide a reasonable explanation for up-stroke detection solely in terms of the biophysics of bursting. I-cells on the other hand receive indirect inhibitory input via granule cells (Carr et al., 1982), the function of which is to invert the signals from electroreceptor afferents (Fig. 4C). Therefore a down-stroke of the external stimulus corresponds to the falling phase of the inhibitory current to the pyramidal cell and thus to the positive slope of the total synaptic drive. We simulated this scenario by inverting the sign of the stimulus to inhibit the soma in the presence of tonic excitatory drive to the dendrites. Figure 4, A3 and D, shows that bursts occur selectively on the down-strokes of the external stimulus. Thus, by inverting the stimulus it becomes possible for bursting cells to detect the negative slope of external stimuli. Therefore within an appropriate network, bursts can be used for bidirectional slope detection.

View larger version (30K): [in this window] [in a new window]   Figure 4.   Bidirectional slope detection. A1, Up-stroke detection. Somatic voltage recording from the E-cell injected with random current (see below) for 3 sec (mean, 1.4; SD, 0.5 nA). Bursts and single spikes were identified using a 10 msec ISI threshold. Arrows show that most bursts occur on positive input slopes; isolated spikes (marked with dots) can occur on both up- and down-strokes. A2, Random input (with cutoff at 5 Hz). A3, Down-stroke detection. Simulations with inverted input like at the I-type cells of electric fish. Dendritic compartment receives 2 nA constant input. Somatic input receives the random input with a mean of 3.5 nA and SD of 7.5 nA. Arrows show that now bursts occur on negative slopes. B, Histogram of input slopes at burst spike times for E-type pyramidal cells. C, Structure of the model. Excitation is applied to dendrites of the E-cell. An inhibitory interneuron can invert the signal in which case inhibition is applied to the soma and tonic excitatory drive to the dendrites of the I-cell. The electrosensory lateral line-lobe of the weakly electric fish, which is discussed in Results, has similar anatomical circuits (Carr et al., 1982). D, Histogram of input slopes at burst spike times for I-type pyramidal cells.

Burst duration coding

We next asked if bursts only mark the occurrence of slopes or whether they are also capable of signaling the magnitude of the slopes as well. Graded signaling might occur by variations in burst duration (Perkel and Bullock, 1968). Although it is often said that bursts are stereotyped, this is not supported by the evidence (Bair et al., 1994; Reinagel et al., 1999), which reveals considerable variability in the number of spikes per burst (typically two to six). Could the number of spikes per burst and/or burst duration vary with the magnitude of the slope? Figure 5A shows that bursts with different number of spikes signal distinct input slopes. Note that this mode of coding is discrete (spikes per burst). This discrete code can be further refined by also considering the time duration of bursts (Fig. 5D). Using signal detection theory we quantified how well bursts containing different number of spikes signal the slope and compared this to how well they signal the amplitude of the input. Figure 5C shows the ROC curves for slope and amplitude signaling with bursts. This measure shows that for any given false alarm rate, slope is signaled with higher accuracy than amplitude (Fig. 5C). To determine whether our conclusions are valid for different types of input we calculated a discriminability index (Gabbiani and Koch, 1998) for inputs with different mean stimulus strengths. The discriminability index is defined as the area under the ROC curve. A value of 0.5 represents chance level, whereas 1 represents perfect discriminability. As shown in Figure 5D, burst duration signals the amplitude of the input poorly, but encodes input slope with high fidelity over a large range of stimulus strengths.

View larger version (35K): [in this window] [in a new window]   Figure 5.   Burst duration codes for input slope. A, Distributions of input slopes for bursts of different durations. B2 represents two spike bursts, B3 three spike bursts, and so on. Different types of bursts signal distinct slopes with little overlap. B, Distributions of input amplitudes for bursts of different durations. The distributions overlap substantially. C, ROC curves show the rate of correct detection for a given false alarm rate. Diagonal represents chance level. Curves show discriminability of two and three spike burst distributions in A and B. Solid curve, Slope signaling; dashed curve, amplitude signaling. D, Discriminability index (area under ROC curve) between two and three spike bursts for slope (solid) and amplitude (dashed) over a range of random stimuli with different mean strengths.

Theoretical basis for slope detection and burst duration coding

The slope sensitivity of bursts may be qualitatively understood as follows. A bursting neuron obeys the current balance equation:

(1)

Without active currents (I_HH and I_BURST) on the right side of the equation, C_mdV/dt = I_INPUT dictates that the change of membrane voltage be proportional to the input current. With the addition of spiking currents, I_HH, voltage change is translated into a spike frequency code whereby the instantaneous frequency becomes a function of input amplitude (Fig. 6A). In a bursting neuron there are additional intrinsic currents (I_BURST). At the onset of a burst, I_BURST turns on. When these currents dominate (I_BURST > I_INPUT) (Fig. 3C), spiking becomes less dependent on input amplitude (Fig. 6B). To explain how this leads to slope detection, we base our analysis on the kinetics of the currents defined as I_BURST. In our model, I_BURST is made up of I_KS and I_NaP. However, to make our argument more general, we will only use two properties of these currents that are expected to be present in a large class of biophysical mechanisms able to generate intrinsic bursting. (The argument also assumes that the input current is changing much slower than the duration of action potentials.) One property is the fast positive feedback seen in Figure 1B and generated in our model by I_NaP. The other property is the slower negative feedback shown in Figure 1C, which here is generated by I_KS. Now consider how I_BURST results in positive slope detection. When I_INPUT is increasing, it can rapidly activate I_NaP, which becomes the dominant current. Then the slow outward current catches up and shuts off the burst. This generates a burst on positive I_INPUT slopes. During the downward swing of I_INPUT, the slow outward current, I_KS is already fully activated while the input current is decreasing, and this makes it unlikely that a burst will be triggered when the slope is negative.

View larger version (44K): [in this window] [in a new window]   Figure 6.   Bursting neurons encode the slope, whereas tonically spiking neurons encode the stimulus amplitude. A, Typical frequency input relationship for a tonically spiking neuron (only type 1 Hodgkin-Huxley currents). The stimulus throughout the figure is the same as in Figure 2a. The instantaneous frequency is shown, which was calculated as the reciprocal of an interspike interval. B, Instantaneous frequency input relationship for the bursting model shows no clear amplitude coding. C, For a tonically spiking neuron, frequency does not encode input slope. D, Burst duration is a function of input slope. Steps are caused by the additions of further spikes to bursts; each cluster contains a given spike count.

Why should burst duration depend on the magnitude of the slope? Let us consider the condition for burst termination to express burst duration as a function of input parameters. Bursts terminate (Fig. 2C) when the net slow currents (I_INPUT  I_BURST) that drive bursting fall below spike threshold, I_th (Koch et al., 1995). Recall that I_BURST = I_KS  I_NaP. Because I_NaP is approximately constant during a burst (Fig. 2C), we will use its average value during a burst, A_NaP. Assume that I_INPUT and I_KS change linearly in time during a burst. Then, I_INPUT(t) = A_INPUT + S_INPUTt and I_KS(t) = A_KS + S_KSt, where A_INPUT and A_KS are the values of I_KS and I_INPUT at the onset of a burst, t = 0. S_INPUT is the slope of the input current, whereas S_KS is the mean slope of I_KS during a burst.

Now we consider how the mean activation of I_KS depends on the input. I_KS is an adaptation current providing proportional negative feedback (Wang, 1998). Hence, A_KS tends to track A_INPUT. This occurs because I_KS is elevated by each spike and decays slowly in time, and the average spike rate over a longer interval is proportional to the mean input. Thus, in this regime, A_KS ~ A_INPUT. From our simulations we found that  = 0.9 (see Fig. 8B), but this constant is expected to change with the kinetics of I_KS as well as the properties of the input.

Recall that bursts terminate (t = T_BURST) when I_INPUT  I_BURST falls below I_th. This can be expressed as I_INPUT + I_NaP  I_KS  A_INPUT(1  ) + A_NaP + (S_INPUT  S_KS)T_BURST  I_th. From this we can now express burst duration:

(2)

This equation can also be obtained by a fast-slow variable mathematical analysis of burst generation (see Appendix). This relationship shows that burst duration depends only weakly on the input amplitude, A_INPUT. By contrast, burst duration increases strongly with input slope, S_INPUT. This relationship between burst duration, T_BURST, and input slope, S_INPUT, captures the mean simulation data (Fig. 6D). Note that Figure 6D also shows clusters representing bursts of a given spike count, which is not accounted for by Equation 2. Thus, for the kinds of burst we have analyzed we are able to provide an explanation for why bursts have positive slope selectivity and why burst duration varies with the magnitude of the slope.

Slope detection with other bursting neurons

In our previous simulations we showed that for a particular type of bursting mechanism involving I_NaP and I_KS, both the timing of bursts as well as the burst duration depend on the input slope. As the analysis in the previous section suggests, this property of bursts is more general. Figure 7A shows that similar burst coding occurs when the bursting mechanism involves high-threshold calcium and calcium-dependent potassium currents (Wang, 1998).

View larger version (26K): [in this window] [in a new window]   Figure 7.   Other bursters also signal slope. A, Slope detection in a neuron model with a calcium-dependent bursting mechanism. The stimulus is random up to a cutoff frequency of 5 Hz (mean 0.5; SD 1.1 µA/cm2). The panel on the right shows the derivative of the stimulus at the onset of the burst as a function of burst duration. Error bars indicate SD. B, Chattering neuron model. Note that chattering cells fire faster bursts and recover more rapidly from after-hyperpolarization, and therefore we used random input with 15 Hz frequency cutoff (mean 2.2; SD 6.5 µA/cm2).

Another class of neurons, the chattering cells located in the superficial layers of visual cortex, can fire brief bursts at very high frequencies (Gray and McCormick, 1996). These bursts are thought to be generated by an interplay between sodium and potassium conductances (Wang, 1999; Brumberg et al., 2000). We used a previously published model of chattering cells (Wang, 1999) to examine the coding properties of bursts. In agreement with our previous findings, the chattering neuron model also fired bursts at the rising edges of the stimulus with a duration proportional the input slope (Fig. 7B).

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We have examined the input-output relationship of a class of bursting neurons using a standard model of bursting pyramidal cells. We first examined the timing of bursts during sinusoidal current injections and found that bursts tend to occur on the rising edge of the input (Fig. 2B) at low frequencies (1-10 Hz) as reported previously (Kamondi et al., 1998; Smith et al., 2000). Next, we examined this slope detection property with more naturalistic current injections. Our simulations showed that bursts are preferentially triggered by positive input slopes (Fig. 3). Single spikes on the other hand showed no slope selectivity at these frequencies. Using random inputs with different, low-frequency components, we found that bursts occur at a small delay (up to 20 msec) with respect to the maximal stimulus slope at higher frequencies (>10 Hz). The preferred range of frequency selectivity is primarily determined by the activation time constant of the slow potassium conductance responsible for terminating bursts. Therefore the burst-mediated mechanism for slope detection is robust within a reasonable range of input statistics.

Is there experimental evidence indicating that bursts signal slope? Studies of weakly electric fish reported that pyramidal neurons detect temporal changes (slope) in the self-generated electric fields (Bastian, 1981; Heiligenberg, 1991). Recent work on these pyramidal cells has shown that bursts are particularly reliable at signaling stimulus slope (Gabbiani et al., 1996; Metzner et al., 1998). However, the origin of this selectivity has been unclear. Gabbiani et al. (1996) pointed out that this selectivity could not be explained with a simple integrate-and-fire mechanism based on their recordings of the membrane potential. Our results show that slope selectivity can result directly from the biophysics of a burst generation process. Interestingly, slope detection is accomplished by two different groups of cells in the electrosensory lateral-line lobe (Fig. 4C): E-cells that signal the rising and I-cells that signal the falling edge of external stimuli (Bastian, 1981; Shumway and Maler, 1989; Gabbiani et al., 1996). Whereas E-cells receive excitatory input directly from the electroreceptor afferents, I-cells receive indirect stimulus input via inhibitory granule cells (Carr et al., 1982; Berman and Maler, 1998). Using this principle, we showed that by inverting the stimulus to inhibit the neuron while tonically depolarizing it, bursts occur on the negative slopes. Therefore the example of the electric fish shows how slope detection by bursts can be adapted for bidirectional signaling (Fig. 4) and used in a behaviorally important computation (Maler, 1996; Gabbiani and Metzner, 1999).

We have also found several other reports in the literature supporting the notion of slope detection by bursts. A particularly clear example is provided by relay cells of the thalamus. When tonically hyperpolarized, these cells generate bursts that occur exclusively on the rising edge of sinusoidal input current both in vitro in slice (Smith et al., 2000) as well as in vivo in anesthetized cats (Guido et al., 1992). An additional example of slope detection by bursts occurs in the lobster stomatogastric nervous system, where in its bursting mode, a sensory neuron encodes the positive slope of muscle stretch (J. T. Birmingham, personal communication).

Diversity of bursting mechanisms

While burst firing can result from a multiplicity of ionic mechanisms there are key properties that define a large class of bursting neurons (Rinzel, 1987; Wang and Rinzel, 1995). First, bursts can outlast the input that triggered them (Fig. 1B). Second, because they are strongly adapting, firing will stop abruptly even if input is sustained (Fig. 1C). These properties of bursts result from ionic conductances that provide fast positive feedback to boost firing rates and slower negative feedback to make bursts strongly adapting. We define any set of current or currents that accomplish this as the burst currents, I_BURST. This can be thought of a module of conductances that perform in a functionally similar manner.

Although we use the I_NaP and I_KS currents to create a "burst module", other conductances can be also used that work together in a similar way. For instance, there is evidence that a high-threshold calcium current, I_CaL, and calcium-dependent potassium current, I_AHP, generate bursting (Fig. 7A) in pyramidal cells (Wong et al., 1979; Golding et al., 1999; Magee and Carruth, 1999; Williams and Stuart, 1999). Our results show that these bursting currents can make neurons more sensitive to the slope of the input current than its amplitude (Figs. 2, 3, 7). When I_BURST has the functional form defined above, we showed that both the timing as well as the duration of bursts will depend on input slope.

Other classes of neurons can have different types of bursting mechanisms. For instance, low-threshold bursts in the thalamus require a preceding hyperpolarization to deinactivate T-type calcium channels (Jahnsen and Llinás, 1984; Zhan et al., 1999), and the inactivation of these channels is primarily responsible for burst termination (Wang et al., 1991). Therefore, although our arguments still suggest that the timing of bursts will be controlled by input slope, as observed previously (Guido et al., 1992; Smith et al., 2000), the duration of bursts may encode other stimulus variables such as the length of preburst silent period. Because neurons are equipped with a rich repertoire of ionic mechanisms that result in bursting (Epstein and Marder, 1990; Wang and Rinzel, 1995), it will be interesting to compare the rules by which different types of bursters transform their inputs.

Spatial control of bursting

Although our results clearly show that bursts can signal slope, the temporal aspects of input may not be the only determinants of whether a burst occurs. Indeed, there is evidence that burst generation in pyramidal cells can additionally be gated by top-down, feedback signals from higher centers that terminate on the distal apical dendrites (Cauller et al., 1998; Larkum et al., 1999; Bastian and Nguyenkim 2001). Therefore, it is possible that cells that fire bursts in response to the temporal properties of the input (slope) may only do so when a certain spatial configuration of inputs is present (Larkum et al., 1999; Bastian and Nguyenkim, 2001). According to this notion, the propensity to burst is enhanced by excitatory input onto the distal dendrites, but whether a burst will occur will depend on the temporal properties of more proximal inputs. In fact, the models we analyzed incorporate burst generating conductances that are located in the dendritic compartment. Although this is not strictly necessary for slope-detection, it can lay the ground for further work to understand the interplay between spatial and temporal factors in the control of bursting (Kepecs and Wang, 2000).

Burst duration coding

Our results also revealed a new aspect of burst signaling: burst duration coding. We found that the differences in burst duration, specifically the spike count (Fig. 5A) and the time duration (Fig. 6D) signal the magnitude of the input slope in a graded manner. Burst duration is a potentially significant neural code because it allows graded information to be transmitted in <20 msec. For example, if action potential frequency is 250 Hz during a burst, bursts containing 2-6 spikes would have durations ranging from 4 to 20 msec (Guido et al., 1992). This gradation is only possible because firing rate has been pushed to its upper limit because of the bursting mechanism. Because multiple spikes can occur in a short period, individual cells can rapidly provide statistically strong information about the input. Given this power of burst duration coding, it will be important to investigate its occurrence in different types of neurons. There is some indication that it occurs in complex cells of the visual cortex, where the number of spikes per burst is modulated by stimulus orientation (DeBusk et al., 1997) and preliminary evidence shows burst duration coding in the lateral geniculate nucleus (Kepecs et al., 2001).

We have addressed a fundamental aspect of neuronal signaling, how input to a neuron is transformed into an output spike pattern. The standard view of this process is that stimulus intensity is transformed into a spike frequency code (Adrian, 1932; Shadlen and Newsome, 1994). At the level of individual neurons this intensity-to-frequency transduction relies on the spiking currents, as described by Hodgkin and Huxley (1952). These spiking currents (I_HH) are present in most neurons and generate action potentials with a frequency proportional to the magnitude of current input. The spiking process can be captured by the integrate-and-fire model (Lapicque, 1907; Tuckwell, 1988; Kistler et al., 1997), which has provided a useful framework to understand the response properties of many types of neurons (Reich et al., 1997; Shadlen and Newsome, 1998; Binder et al., 1999).

Our study emphasizes that as neurons are endowed with a richer repertoire of voltage-gated conductances, their input-output relationship becomes more complex, and their computational capabilities change. In particular, these intrinsic conductances can determine whether a cell will burst or fire regular spike trains. As we show here, some bursting neurons can perform a differentiate and burst computation that is fundamentally different from the integrate and fire function of simple spiking cells. The ability of these bursting cells to signal slope (Fig. 6) can be used to make behaviorally important computations and may underlie a burst duration code useful for rapid information transfer.

	FOOTNOTES

Received Jan. 30, 2002; revised July 16, 2002; accepted July 24, 2002.

This work was supported by grants from the National Institutes of Health, W. M. Keck Foundation, and the Alfred P. Sloan Foundation. We thank Larry Abbott, Eve Marder, and Sridhar Raghavachari for comments on a previous version of this manuscript and Jeremy Caplan, Enrique Garibay, and Sridhar Raghavachari for useful discussions.

Correspondence should be addressed to Xiao-Jing Wang, Volen Center for Complex Systems, MS 013, Brandeis University, 415 South Street, Waltham, MA 02454-9110. E-mail: xjwang{at}brandeis.edu.

	APPENDIX

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Bursts can be analyzed mathematically with the "fast-slow variable dissection method" (Rinzel, 1987; Rinzel and Ermentrout, 1989). The method is based on the idea that different components of bursting act on very different time scales and can therefore be analyzed separately. In our case, the potassium current I_KS and the input current I_INPUT vary much more slowly than the spike-generating dynamics. The slow and fast membrane processes can be therefore separated by first considering how the behavior of the fast spiking subsystem depends on the slow, I(t) = I_KS  I_INPUT, as if the latter were a fixed parameter. In a second step, the slow time course of I(t) is considered, which then predicts how the burst firing pattern evolves in time. This analysis clarifies the assumptions made in the previous derivation and in particular eliminates the need to assume that bursts terminate at a given current threshold, I_th.

Figure 8A shows a bifurcation diagram (Kepecs and Wang, 2000), where the fast system is represented by the neuronal firing frequency as a function of I = I_KS  I_INPUT. The dynamics of the slow system is represented by the arrows. At a critical value, I₁, bursting commences with maximal frequency. During a burst, I_KS increases slowly but at a higher rate than I_INPUT, so that I increases, and the firing frequency monotonically decreases. Using a linear approximation, we can express this relationship as f(I) = f₀ where f₀ = f(I₁) is the frequency at the onset of bursting. At the beginning of the burst, t = 0 and I = I₁. From Figure 2C we observe that I_KS increases in an approximately linear manner during a burst: I_KS = A_KS + S_KSt, and we rewrite the input current similarly as I_INPUT = A_INPUT + S_INPUTt. Now, I₁ = A_KS  A_INPUT. As an adaptation current the mean I_KS tends to track the mean input (Wang, 1998), A_KS = C_KS + A_INPUT. From our simulations we find that  = 0.9 (Fig. 8B). Thus, I₁ = (  1)A_INPUT + C_KS. We can now write I(t) = I₁ + (S_KS  S_INPUT)t. Using these relationships we can write the time evolution of frequency as f(t) = f₀ A burst terminates (t = T) when its frequency goes to zero: f(T) = 0. From these we can express burst duration, with const₁ = I₂  C_KS and const₂ = S_KS. Thus burst duration increases nonlinearly and dramatically with input slope, S_INPUT , whereas its dependence on the input amplitude, A_INPUT , is weak.

View larger version (16K): [in this window] [in a new window]   Figure 8.   Determinants of bursting. A, Bifurcation diagram of the model showing how spike frequency changes as a function of input. Bursting starts at I1 and ends at a homoclinic bifurcation at I2. Arrows mark the evolution (decrease) of frequency during bursts and the quiescent period in between bursts. B, Mean IKS tracks IINPUT. Dots show data points from individual bursts. Both the input current as well as the current from the slow potassium channels were measured at the onset of bursting. The best fit line has a slope of 0.9.

Our argument still holds if the dependence of the firing rate f on I is not linear, e.g., is instead a quadratic function near the saddle node bifurcation at I₂ (Rinzel, 1987; Rinzel and Ermentrout, 1989).

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TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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