Abstract
Comprehending how the brain functions requires an understanding of the dynamics of neuronal assemblies. Previous work used a mean-field reduction method to determine the collective dynamics of a large heterogeneous network of uniformly and globally coupled theta neurons, which are a canonical formulation of Type I neurons. However, in modeling neuronal networks, it is unreasonable to assume that the coupling strength between every pair of neurons is identical. The goal in the present work is to analytically examine the collective macroscopic behavior of a network of theta neurons that is more realistic in that it includes heterogeneity in the coupling strength as well as in neuronal excitability. We consider the occurrence of dynamical structures that give rise to complicated dynamics via bifurcations of macroscopic collective quantities, concentrating on two biophysically relevant cases: (1) predominantly excitable neurons with mostly excitatory connections, and (2) predominantly spiking neurons with inhibitory connections. We find that increasing the synaptic diversity moves these dynamical structures to distant extremes of parameter space, leaving simple collective equilibrium states in the physiologically relevant region. We also study the node vs. focus nature of stable macroscopic equilibrium solutions and discuss our results in the context of recent literature.
Highlights
In 1949, Hebb (1949) proposed that cell assemblies are the true functional unit of the nervous system
To derive the reduced dynamical system for our network, we follow the methods of Ott and Antonsen (2008, 2009), Marvel et al (2009), and Luke et al (2013), but include heterogeneity in the coupling strength according to Equation (3)
We constructed a large network of theta neurons that included diversity in the excitability parameters as well as connections with diversity in their coupling strengths
Summary
In 1949, Hebb (1949) proposed that cell assemblies are the true functional unit of the nervous system. The cerebral cortex contains networks of neuronal assemblies that comprise a large number of interacting neurons (Harris, 2005; Sporns et al, 2005). Individual neuronal assemblies organize via transient synchronization to generate collective behavior that is critical to communication between the neuronal assemblies themselves. In developing an analytical understanding of the behavior of large neuronal assemblies, it is prohibitively challenging to use realistic models of actual neurons. A model can be considered canonical for a family of models if a continuous change of variables can transform any instance of that family into the canonical model. Such a model is advantageous due to its universality since any behavior exhibited by the canonical
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