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Rate Models for Conductance-Based Cortical Neuronal Networks

orens{at}fiz.huji.ac.il, Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel, and Center for Neural Computation, Hebrew University, Jerusalem 91904, Israel

David Hansel

david.hansel{at}biomedicale.univ-paris5.fr, Laboratoire de Neurophysique et Physiologie du Système Moteur, Université René Descartes, 75270 Paris cedex 06, Paris, France, and Center for Neural Computation, Hebrew University, Jerusalem 91904, Israel

Haim Sompolinsky

haim{at}fiz.huji.ac.il, Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel, and Center for Neural Computation, Hebrew University, Jerusalem 91904, Israel

Population rate models provide powerful tools for investigating the principles that underlie the cooperative function of large neuronal systems. However, biophysical interpretations of these models have been ambiguous. Hence, their applicability to real neuronal systems and their experimental validation have been severely limited. In this work, we show that conductance-based models of large cortical neuronal networks can be described by simplified rate models, provided that the network state does not possess a high degree of synchrony. We first derive a precise mapping between the parameters of the rate equations and those of the conductance-based network models for time-independent inputs. This mapping is based on the assumption that the effect of increasing the cell's input conductance on its f-I curve is mainly subtractive. This assumption is confirmed by a single compartment Hodgkin-Huxley type model with a transient potassium A-current. This approach is applied to the study of a network model of a hypercolumn in primary visual cortex. We also explore extensions of the rate model to the dynamic domain by studying the firing-rate response of our conductance-based neuron to time-dependent noisy inputs. We show that the dynamics of this response can be approximated by a time-dependent second-order differential equation. This phenomenological single-cell rate model is used to calculate the response of a conductance-based network to time-dependent inputs.

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