Neural and Cognitive Modeling with Networks of Leaky Integrator Units

Abstract

Summary. After reviewing several physiological findings on oscillations in the electroencephalogram (EEG) and their possible explanations by dynamical modeling, we present neural networks consisting of leaky integrator units as an universal paradigm for neural and cognitive modeling. In contrast to standard recurrent neural networks, leaky integrator units are described by ordinary differential equations living in continuous time. We present an algorithm to train the temporal behavior of leaky integrator networks by generalized backpropagation and discuss their physiological relevance. Eventually, we show how leaky integrator units can be used to build oscillators which may serve as models of brain oscillations and cognitive processing.

Existence and properties of solutions for neural field equations

“…Modeling neurodynamics has a long tradition in mathematical biology and computational neuroscience, starting with the study of simple neuron models and the theory of neural networks in the 1940ies [1][2][3][4][5][6][7][8]. One particular neuron model with certain physiological significance is the leaky integrator unit [2,3,[5][6][7][8] described by the ODEs…”

“…One particular neuron model with certain physiological significance is the leaky integrator unit [2,3,[5][6][7][8] described by the ODEs…”

On the Electrodynamics of Neural Networks

“…the apical dendritic tree and E E ij is the excitatory reversal potential of the synapse j → i. We can conveniently express γ(t) through the spike rate [15,33] a…”

Mean-field dynamics of a random neural network with noise

“…This is typically resolved by invoking the mean-field approach to describe the coarse-grained activity and interactions of neural populations. Given the often used assumption on population homogeneity, the approach is from biological view most appropriate for intermediate-scale (mesoscopic) assemblies, such as cortical columns [5,6]. The latter assemblies incorporate on one hand a sufficiently large number of neurons for the averaging effects to occur, but on the other hand, are small enough to support the homogeneity assumption.…”

“…The mean field approach has so far been implemented to network structures as well as spatially extended neural systems, with the pertaining models classified as activitybased or voltage-based depending on the type of the state variable [6,7]. The seminal works of Wilson and Cowan [8,9], as well as Amari [10], employed the heuristic continuum limit, providing the description of the temporal coarse-grained dynamics in neural fields.…”