Elsevier

Neural Networks

Volume 12, Issues 7–8, October–November 1999, Pages 1181-1190
Neural Networks

Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons

https://doi.org/10.1016/S0893-6080(99)00053-2Get rights and content

Abstract

There has been controversy over whether the standard neuro-spiking models are consistent with the irregular spiking of cortical neurons. In a previous study, we proposed examining this consistency on the basis of the high-order statistics of the inter-spike intervals (ISIs), as represented by the coefficient of variation and the skewness coefficient. In that study we found that a leaky integrate-and-fire model incorporating the assumption of temporally uncorrelated inputs is not able to account for the spiking data recorded from a monkey prefrontal cortex. In the present paper, we attempt to revise the neuro-spiking model so as to make it consistent with the biological data. Here we consider the correlation coefficient of consecutive ISIs, which was ignored in previous studies. Considering three statistical coefficients, we conclude that the leaky integrate-and-fire model with temporally correlated inputs does account for the biological data. The correlation time scale of the inputs needed to explain the biological statistics is found to be on the order of 100 ms. We discuss possible origins of this input correlation.

Introduction

It is known that a cortical neuron in vivo generates irregular spike sequences, while a neuron under a constant current injection in vitro generates regular spike sequences (Kaneko et al., 1995, Thomson and Deuchars, 1997). This apparent disagreement was not considered carefully until Softky and Koch (1993) pointed out the difficulty involving standard neuro-spiking models in reproducing spiking irregularity. Shadlen and Newsome, 1994, Shadlen and Newsome, 1998 took up this issue and pointed out that a standard leaky integrate-and-fire (LIF) model can reproduce the spiking irregularity if the inhibition is balanced with the excitation. Since their investigation, there have been several studies concerning the manner in which balanced inhibition is brought about naturally in model networks (Amit and Brunel, 1997, Tsodyks and Sejnowski, 1995, van Vreeswijk and Sompolinsky, 1996, van Vreeswijk and Sompolinsky, 1998). To our knowledge, however, the original problem of whether standard neuro-spiking models are consistent with the biological spiking irregularity is not yet solved.

We attempted in a previous study to determine whether a standard neuro-spiking model is consistent with the biological spiking data by considering higher order statistics in addition to the measure of spiking irregularity (Shinomoto and Sakai, 1998, Shinomoto et al., 1999). The consistency of a neuro-spiking model with biological spiking data was examined by the coefficient of variation ‘CV’ and the skewness coefficient ‘SK’, which are, respectively, a measure of the spiking irregularity and a measure of the asymmetry of the inter-spike interval (ISI) distribution. It was found that the LIF model with the assumption of temporally uncorrelated inputs is unable to account for spiking data recorded from a monkey prefrontal cortex.

Interpreting this fact as implying the need to revise neuro-spiking models, we attempt in the present study to revise this model so as to reconcile it with the biological data. A neuro-spiking model can be decomposed into two parts, a neuron itself (the spiking mechanism of a single neuron) and input signals to the neuron (statistical characteristics of incoming inputs). The newest single neuron models contain a number of variables which interact with each other in a complex manner (Yamada, Koch & Adams, 1989). Furthermore, it has become evident that input signals can operate on a single neuron in a complicated non-linear fashion by means of active dendrites and other influences (Koch, 1997). It is tempting to consider these non-linear mechanisms as representing some of the ingredients responsible for the apparently irregular spike sequences. It is nevertheless true that the simple leaky integration mechanism lies at the base of neuro-spiking mechanisms. It would therefore be worthwhile to examine whether classical single neuron models, such as the simple LIF model, can reproduce biological spiking statistics, by appropriately choosing the statistical nature of the incoming inputs.

For the purpose of detailed data examination, we employ here three statistical coefficients, adding the correlation coefficient of consecutive ISIs ‘COR’, to the previously employed statistical coefficients, CV and SK. The correlation coefficient COR is expected to vanish in the renewal process, in which consecutive ISIs are generated independently with each other. We find that fairly large percentage of the biological data exhibit anomalously large COR values in comparison with the possible range of deviation for the renewal process. All the standard single neuron models incorporating the assumption of temporally uncorrelated inputs belong to the renewal process, and are therefore rejected as reasonable neuro-spiking models.

Alternatively, we can discard the data with large |COR| values, regarding them as being generated by some non-renewal processes, and examine the remaining population in the CV–SK plane to see whether those data are consistent with the Ornstein–Uhlenbeck process (OUP) or other standard neuro-spiking models. It turns out, however, that the remaining data rather exhibit larger inconsistency with the standard neuro-spiking models, as we select data with smaller |COR| values.

Finally we examine the temporally correlated inputs. We then found that even a simple LIF model does behave consistently with the biological data if it is subject to temporally correlated inputs. Adopting the LIF model with temporally correlated inputs as a neuro-spiking model, we estimate the range of the correlation time scale of the inputs needed to explain the biological data. The time scale turns out to be of the same order as the mean ISI, which corresponds to approximately 100 ms for our data. We discuss possible origins of this input correlation.

While we were preparing the present manuscript, we came across an interesting experimental paper by Stevens and Zador (1998) whose conclusion includes some of those reached in the present study. They measured spiking irregularity from spiking data of biological neurons which were injected with temporally fluctuating currents. They concluded that independent inputs to a neuron do not account for the ISI variability but that this variability can be explained by transiently synchronous inputs to a neuron.

The first half of their conclusion does not agree with ours: For the case of uncorrelated inputs, they did not observe irregular spike sequences in their experiment, whereas we could in fact generate spike sequences with large CV values in our numerical simulation. This disagreement is presumably not due to an essential disagreement between the biological reality and the model but rather, due to some practical difficulty accompanied with experiment. The second half of their conclusion is similar to ours: They found that ‘transiently synchronous inputs’ to a neuron can result in a large CV value. Such ‘transiently synchronous inputs’ can be interpreted as the ‘temporally correlated inputs’ in our neuro-spiking model. Their conclusion has some similarity to ours in the sense that transiently synchronous inputs (or temporally correlated inputs) are necessary to explain some aspects of spiking statistics. However, we wish to stress again that a large CV value alone can be realized even with the uncorrelated inputs. We would like to propose examining spiking data by considering the three statistical coefficients CV, SK and COR.

Section snippets

Biological spiking data and the statistical coefficients

We examine neuro-spiking models through comparison with biological spiking data obtained from the delay response task experiment carried out by Funahashi, whose task paradigm is identical to one of the varieties presented in Funahashi et al., 1989, Goldman-Rakic et al., 1990. In the experiment, a rhesus monkey is required to make a specific saccade eye movement in response to a visual cue stimulus which is presented in advance to a 3 s delay period during which cue stimulus is absent. Neurons in

Leaky integrate-and-fire model

The leaky integration mechanism is the common base of neuro-spiking mechanisms. Incoming spiking signals to a neuron pull its membrane potential either up or down, depending on the characteristics of the arrival synaptic junctions. Between inputs, the membrane potential of a neuron is always decaying toward the resting level. If the membrane potential exceeds a certain threshold, the neuron fires and emits a spike. The potential then swiftly returns to a near-resting level.

There are many

Temporally uncorrelated (white) inputs

Throughout this paper, we consider only the case in which the ‘(inputs)’ term corresponds to a stationary stochastic process. Stationary stochastic processes can be classified into two groups, temporally uncorrelated and temporally correlated. These are known respectively as ‘white’ and ‘colored’, in analogy to the power spectrum of light. First, we consider single neuron models that are subject to temporally uncorrelated (white) inputs, and we will then move to the LIF model which is subject

Temporally correlated (colored) inputs

Any neuro-spiking model consisting of a one-dimensional first-order differential equation with temporally uncorrelated inputs is a renewal process, the large COR values exhibited by biological data cannot be accounted for by a renewal process. We wish to look for a neuro-spiking model which can reproduce the wide distribution of the data we have analyzed in the space of the statistical coefficients (CV,SK,COR). The LIF model with temporally correlated inputs is not a type of renewal process. We

Discussion

The LIF model with temporally correlated inputs turns out to account for the biological data widely distributed in the space of the three statistical coefficients CV, SK and COR. We can also estimate the reasonable range of the correlation time scale of the inputs to a neuron. The time scale was found to be on the same order as the mean ISI, which is typically 100 ms for our data. Therefore the input correlation time scale thus estimated is distinctly larger than the membrane decay time

Acknowledgements

This study is supported in part by a Grant-in-Aid for Scientific Research on Priority Areas (no. 08279103) to S. Shinomoto by the Ministry of Education, Science, Sports and Culture of Japan. We are grateful to Hiroshi Fujii for information on some recent papers related to our study.

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