I have been engaged in the study of how variability can arise in neuronal firing. In particular, I have focused on the variability that would be generated through the recurrent connections within a network. Most of the past work on the dynamics of interacting neurons or oscillators have focused on the infinite system size limit where fluctuations due to the connections do not appear. However, many biological and neural networks are large but finite sized. The dynamics of such networks are not well understood. With post doctoral fellow, Hedi Soula, we studied the population dynamics of a finite number of stochastically firing neurons. We were able to analytically deduce statistical properties of the network such as the mean and variance of the firing rate. In particular, we showed for a finite population of neurons, the mean firing rate can be approximated using mean field theory but the variance cannot. Fellow Michael Buice and I examined the dynamics of a large but finite size network of globally connected oscillators. The model is the weak coupling limit of a mutually connected network of neurons that have a tendency to synchronize due to the connections. We showed that ideas from the kinetic theory of gases and plasmas could be applied to analyze the fluctuations and correlations due to system size effects. In particular, we showed that finite population size could stabilize the marginal asynchronous mode. This had been an open problem for twenty years. Recently, we have developed a scheme to generalize population rate equations to account for correlations. The approach shows how a moment hierarchy can be generated from an underlying Master equation. We are now generalizing the result to deterministic systems. Fellow Shashaank Vattikuti and I have been exploring how genetic and anatomical perturbations can give rise to autism spectrum disorder traits. In particular, we used a local cortical circuit at the sub-millimetre level as a bridge between genotype and phenotype. We have found that synaptic imbalance and changes in the minicolumn structure of cortex can effect performance in visual saccade tasks that match experiments. The model also makes predictions for possible pharmacological therapeutics. Fellow Jeffrey Seely and I have illuminated the role of mutual inhibition in binocular rivalry.

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Vattikuti, Shashaank; Thangaraj, Phyllis; Xie, Hua W et al. (2016) Canonical Cortical Circuit Model Explains Rivalry, Intermittent Rivalry, and Rivalry Memory. PLoS Comput Biol 12:e1004903
Chow, Carson C; Buice, Michael A (2015) Path integral methods for stochastic differential equations. J Math Neurosci 5:8
Buice, Michael A; Chow, Carson C (2013) Dynamic finite size effects in spiking neural networks. PLoS Comput Biol 9:e1002872
Buice, Michael A; Chow, Carson C (2013) Generalized activity equations for spiking neural network dynamics. Front Comput Neurosci 7:162
Buice, Michael A; Chow, Carson C (2011) Effective stochastic behavior in dynamical systems with incomplete information. Phys Rev E Stat Nonlin Soft Matter Phys 84:051120
Seely, Jeffrey; Chow, Carson C (2011) Role of mutual inhibition in binocular rivalry. J Neurophysiol 106:2136-50
Buice, Michael A; Cowan, Jack D; Chow, Carson C (2010) Systematic fluctuation expansion for neural network activity equations. Neural Comput 22:377-426
Vattikuti, Shashaank; Chow, Carson C (2010) A computational model for cerebral cortical dysfunction in autism spectrum disorders. Biol Psychiatry 67:672-8