Abstract
Networks of neurons in the cerebral cortex exhibit a balance between excitation (positive input current) and inhibition (negative input current). Balanced network theory provides a parsimonious mathematical model of this excitatory-inhibitory balance using randomly connected networks of model neurons in which balance is realized as a stable fixed point of network dynamics in the limit of large network size. Balanced network theory reproduces many salient features of cortical network dynamics such as asynchronous-irregular spiking activity. Early studies of balanced networks did not account for the spatial topology of cortical networks. Later works introduced spatial connectivity structure, but were restricted to networks with translationally invariant connectivity structure in which connection probability depends on distance alone and boundaries are assumed to be periodic. Spatial connectivity structure in cortical network does not always satisfy these assumptions. We use the mathematical theory of integral equations to extend the mean-field theory of balanced networks to account for more general dependence of connection probability on the spatial location of pre- and postsynaptic neurons. We compare our mathematical derivations to simulations of large networks of recurrently connected spiking neuron models.
Highlights
Balanced networks [1, 2] offer a parsimonious computational and mathematical model of the asynchronous-irregular spiking activity and excitatory-inhibitory balance that are ubiquitous in cortical neuronal networks [3,4,5,6,7,8,9,10]
We use the mathematical theory of integral equations [21] to extend the mean-field theory of firing rates in balanced networks, permitting a more general dependence of connection probability on the spatial location of pre- and postsynaptic neurons
5 Discussion We extended the mean-field theory of firing rates in balanced networks to account for spatial connectivity structures in which connection probabilities depend on the spatial location of pre- and postsynaptic neurons without the translation invariance assumed in previous work
Summary
Balanced networks [1, 2] offer a parsimonious computational and mathematical model of the asynchronous-irregular spiking activity and excitatory-inhibitory balance that are ubiquitous in cortical neuronal networks [3,4,5,6,7,8,9,10]. We use the mathematical theory of integral equations [21] to extend the mean-field theory of firing rates in balanced networks, permitting a more general dependence of connection probability on the spatial location of pre- and postsynaptic neurons. The mean-field theory of balanced networks offers an alternative approach to analyzing firing rates in which the mapping from synaptic input statistics to rates does not need to be known This theory is developed and applied by analyzing Eq (5) and the integral equations implied by it, which serves as a heuristic approximation to our spiking network model. Note that W may not have an orthonormal basis of eigenfunctions with real eigenvalues even when kernels are spatially symmetric (wab(x, y) = wab(y, x)) because a Hermitian W would require that wei(x, y) = wie(y, x), which is generally not true for excitatory-inhibitory networks. We compare Eqs. (12) and (13) to results from large spiking network simulations
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