Exact differential equation population dynamics for integrate-and-fire neurons

Part of Advances in Neural Information Processing Systems 14 (NIPS 2001)

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Julian Eggert, Berthold Bäuml


In our previous work, integral equation formulations for

Mesoscopical, mathematical descriptions of dynamics of popula(cid:173) tions of spiking neurons are getting increasingly important for the understanding of large-scale processes in the brain using simula(cid:173) tions. population dynamics have been derived for a special type of spik(cid:173) ing neurons. For Integrate- and- Fire type neurons, these formula(cid:173) tions were only approximately correct. Here, we derive a math(cid:173) ematically compact, exact population dynamics formulation for Integrate- and- Fire type neurons. It can be shown quantitatively in simulations that the numerical correspondence with microscop(cid:173) ically modeled neuronal populations is excellent.


Introduction and motivation

The goal of the population dynamics approach is to model the time course of the col(cid:173) lective activity of entire populations of functionally and dynamically similar neurons in a compact way, using a higher descriptionallevel than that of single neurons and spikes. The usual observable at the level of neuronal populations is the population(cid:173) averaged instantaneous firing rate A(t), with A(t)6.t being the number of neurons in the population that release a spike in an interval [t, t+6.t). Population dynamics are formulated in such a way, that they match quantitatively the time course of a given A(t), either gained experimentally or by microscopical, detailed simulation.

At least three main reasons can be formulated which underline the importance of the population dynamics approach for computational neuroscience. First, it enables the simulation of extensive networks involving a massive number of neurons

and connections, which is typically the case when dealing with biologically realistic functional models that go beyond the single neuron level. Second, it increases the analytical understanding of large-scale neuronal dynamics, opening the way towards better control and predictive capabilities when dealing with large networks. Third, it enables a systematic embedding of the numerous neuronal models operating at different descriptional scales into a generalized theoretic framework, explaining the relationships, dependencies and derivations of the respective models.

Early efforts on population dynamics approaches date back as early as 1972, to the work of Wilson and Cowan [8] and Knight [4], which laid the basis for all current population-averaged graded-response models (see e.g. [6] for modeling work using these models). More recently, population-based approaches for spiking neurons were developed, mainly by Gerstner [3, 2] and Knight [5]. In our own previous work [1], we have developed a theoretical framework which enables to systematize and sim(cid:173) ulate a wide range of models for population-based dynamics. It was shown that the equations of the framework produce results that agree quantitatively well with detailed simulations using spiking neurons, so that they can be used for realistic simulations involving networks with large numbers of spiking neurons. Neverthe(cid:173) less, for neuronal populations composed of Integrate-and-Fire (I&F) neurons, this framework was only correct in an approximation. In this paper, we derive the exact population dynamics formulation for I&F neurons. This is achieved by reducing the I&F population dynamics to a point process and by taking advantage of the particular properties of I&F neurons.

2 Background: Integrate-and-Fire dynamics

2.1 Differential form

We start with the standard Integrate- and- Fire (I&F) model in form of the well(cid:173) known differential equation [7]


which describes the dynamics of the membrane potential Vi of a neuron i that is modeled as a single compartment with RC circuit characteristics. The membrane relaxation time is in this case T = RC with R being the membrane resistance and C the membrane capacitance. The resting potential v R est is the stationary potential that is approached in the no-input case. The input arriving from other neurons is described in form of a current ji.

In addition to eq. (1), which describes the integrate part of the I&F model, the neuronal dynamics are completed by a nonlinear step. Every time the membrane potential Vi reaches a fixed threshold () from below, Vi is lowered by a fixed amount Ll > 0, and from the new value of the membrane potential integration according to eq. (1) starts again.

if Vi(t) = () (from below) .


At the same time, it is said that the release of a spike occurred (i.e., the neuron fired), and the time ti = t of this singular event is stored. Here ti indicates the time of the most recent spike. Storing all the last firing times, we gain the sequence of spikes {t{} (spike ordering index j, neuronal index i).