nips nips2001 nips2001-72 nips2001-72-reference knowledge-graph by maker-knowledge-mining
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Author: Julian Eggert, Berthold Bäuml
Abstract: Mesoscopical, mathematical descriptions of dynamics of populations of spiking neurons are getting increasingly important for the understanding of large-scale processes in the brain using simulations. In our previous work, integral equation formulations for population dynamics have been derived for a special type of spiking neurons. For Integrate- and- Fire type neurons , these formulations were only approximately correct. Here, we derive a mathematically compact, exact population dynamics formulation for Integrate- and- Fire type neurons. It can be shown quantitatively in simulations that the numerical correspondence with microscopically modeled neuronal populations is excellent. 1 Introduction and motivation The goal of the population dynamics approach is to model the time course of the collective 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 populationaveraged 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 simulate 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. Nevertheless, 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 2.1 Background: Integrate-and-Fire dynamics Differential form We start with the standard Integrate- and- Fire (I&F;) model in form of the wellknown differential equation [7] (1) 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) . (2) 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). 2.2 Integral form Now we look at the single neuron in a neuronal compound. We assume that the input current contribution ji from presynaptic spiking neurons can be described using the presynaptic spike times tf, a response-function ~ and a connection weight W¡ . ',J ji(t) = Wi ,j ~(t - tf) (3) l: l: j f Integrating the I&F; equation (1) beginning at the last spiking time tT, which determines the initial condition by Vi(ti) = vi(ti - 0) - 6., where vi(ti - 0) is the membrane potential just before the neuron spikes, we get 1 Vi(t) = v Rest + fj(t - t:) + l: Wi ,j l: a(t - t:; t - tf) , j - Vi(t:)) e- S / T (4) f with the refractory function fj(s) = - (v Rest (5) and the alpha-function r ds
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