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This paper introduces a class of stochastic models of interacting neurons with emergent dynamics similar to those seen in local cortical populations. Rigorous results on existence and uniqueness of nonequilibrium steady states are proved. These network models are then compared to very simple reduced models driven by the same mean excitatory and inhibitory currents. Discrepancies in firing rates between network and reduced models are investigated and explained by correlations in spiking, or partial synchronization, working in concert with “nonlinearities” in the time evolution of membrane potentials. The use of simple random walks and their first passage times to simulate fluctuations in neuronal membrane potentials and interspike times is also considered.

Neuroscience models come in a wide range of scales and specificity, from mean-field rate models to large-scale networks of spiking neurons. There are potential trade-offs between simplicity and realism, versatility and computational speed. This paper is about large-scale cortical network models, and the question we address is one of scalability: would scaling down cell density impact a network's ability to reproduce cortical dynamics and function? We investigated this problem using a previously constructed realistic model of the monkey visual cortex that is true to size. Reducing cell density gradually up to 50-fold, we studied changes in model behavior. Size reduction without parameter adjustment was catastrophic. Surprisingly, relatively minor compensation in synaptic weights guided by a theoretical algorithm restored mean firing rates and basic function such as orientation selectivity to models 10-20 times smaller than the real cortex. Not all was normal in the reduced model cortices: intracellular dynamics acquired a character different from that of real neurons, and while the ability to relay feedforward inputs remained intact, reduced models showed signs of deficiency in functions that required dynamical interaction among cortical neurons. These findings are not confined to models of the visual cortex, and modelers should be aware of potential issues that accompany size reduction. Broader implications of this study include the importance of homeostatic maintenance of firing rates, and the functional consequences of feedforward versus recurrent dynamics, ideas that may shed light on other species and on systems suffering cell loss.

This paper offers a theory for the origin of direction selectivity (DS) in the macaque primary visual cortex, V1. DS is essential for the perception of motion and control of pursuit eye movements. In the macaque visual pathway, neurons with DS first appear in V1, in the Simple cell population of the Magnocellular input layer 4Cα. The lateral geniculate nucleus (LGN) cells that project to these cortical neurons, however, are not direction selective. We hypothesize that DS is initiated in feed-forward LGN input, in the summed responses of LGN cells afferent to a cortical cell, and it is achieved through the interplay of 1) different visual response dynamics of ON and OFF LGN cells and 2) the wiring of ON and OFF LGN neurons to cortex. We identify specific temporal differences in the ON/OFF pathways that, together with item 2, produce distinct response time courses in separated subregions; analysis and simulations confirm the efficacy of the mechanisms proposed. To constrain the theory, we present data on Simple cells in layer 4Cα in response to drifting gratings. About half of the cells were found to have high DS, and the DS was broadband in spatial and temporal frequency (SF and TF). The proposed theory includes a complete analysis of how stimulus features such as SF and TF interact with ON/OFF dynamics and LGN-to-cortex wiring to determine the preferred direction and magnitude of DS.

We investigate the macroscopic time evolution and stationary states of a mean field generalized contact process in \\(\\mathbb{R}^d\\). The model is described by a coupled set of nonlinear integral-differential equations. It was inspired by a model of neurons with discrete voltages evolving by a stochastic integrate and fire mechanism. We obtain a complete solution in the spatially uniform case and partial solutions in the general case. The system has one or more fixed points and also traveling wave solutions.

We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice Z d . The states at each lattice site can take values in 0 , … , k . These can be interpreted as neuronal membrane potential, with the state k corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state k corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from 0 to k . The rate at which it climbs is determined by other neurons at state k , coupled to it through a Kac-type potential, of range γ - 1 . The hydrodynamic equations are obtained in the limit γ → 0 . Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered.

This numerical study documents and analyzes emergent spiking behavior in local neuronal populations. Emphasis is given to a phenomenon we call clustering , by which we refer to a tendency of random groups of neurons large and small to spontaneously coordinate their spiking activity in some fashion. Using a sparsely connected network of integrate-and-fire neurons, we demonstrate that spike clustering occurs ubiquitously in both high firing and low firing regimes. As a practical tool for quantifying such spike patterns, we propose a simple scheme with two parameters, one setting the temporal scale and the other the amount of deviation from the mean to be regarded as significant. Viewing population activity as a sequence of events , meaning relatively brief durations of elevated spiking, separated by inter-event times , we observe that background activity tends to give rise to extremely broad distributions of event sizes and inter-event times, while driving a system imposes a certain regularity on its inter-event times, producing a rhythm consistent with broad-band gamma oscillations. We note also that event sizes and inter-event times decorrelate very quickly. Dynamical analyses supported by numerical evidence are offered.

We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice \\(\\mathbb{Z}^d\\). The states at each lattice site can take values in \\(0,\\ldots,k\\). These can be interpreted as neuronal membrane potential, with the state \\(k\\) corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state \\(k\\) corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from \\(0\\) to \\(k\\). The rate at which it climbs is determined by other neurons at state \\(k\\), coupled to it through a Kac-type potential, of range \\(\\gamma^{-1}\\). The hydrodynamic equations are obtained in the limit \\(\\gamma\\rightarrow 0\\). Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered.

This paper offers a new theory for the origin of direction selectivity in the Macaque primary visual cortex, V1. Direction selectivity (DS) is essential for the perception of motion and control of pursuit eye movements. In the Macaque visual pathway, DS neurons first appear in V1, in the Simple cell population of the Magnocellular input layer 4Ca. The LGN cells that project to these cortical neurons, however, are not direction-selective. We hypothesize that DS is initiated in feedforward LGN input, in the summed responses of LGN cells afferent to a cortical cell, and it is achieved through the interplay of (a) different visual response dynamics of ON and OFF LGN cells, and (b) the wiring of ON and OFF LGN neurons to cortex. We identify specific temporal differences in the ON/OFF pathways that together with (b) produce distinct response time-courses in separated subregions; analysis and simulations confirm the efficacy of the mechanisms proposed. To constrain the theory, we present data on Simple cells in layer 4Ca in response to drifting gratings. About half of the cells were found to have high DS, and the DS was broad-band in spatial and temporal frequency (SF and TF). The proposed theory includes a complete analysis of how stimulus features such as SF and TF interact with ON/OFF dynamics and LGN-to-cortex wiring to determine the preferred direction and magnitude of DS. Competing Interest Statement The authors have declared no competing interest.

This thesis is comprised of two parts: first an investigation into the dynamics of generic local populations of neurons, and second the construction of a realistic model of the input layer to the primary visual cortex of the macaque monkey. In part 1, we build a sparsely connected network of a few hundred excitatory and inhibitory integrate and fire neurons meant to be representative of generic local populations throughout cortex. Upon viewing the collective pattern of spikes from neurons in the network, a salient feature appears: one sees the repeated brief coordination of small subgroups of neurons, followed by lulls in firing activity. We refer to these coherent groups of spikes as spike clusters, and the phenomenon in general as spike clustering. Rangan and Young (2013) discovered this phenomenon in their simulations, and we study it more systematically and add new results. A challenge is how to provide a useful definition of spike clusters. We solve this problem, subsequently gathering statistics, identifying the mechanism for spike clustering, and investigating the relationship between model parameters and dynamics. In part 2, we create another integrate and fire network model, this time of a specific location in cortex. Our focus is a small patch of layer 4Cα in macaque V1, containing 9 orientation hypercolumns, located at about 5° eccentricity. We model the LGN connectivity to 4Cα in accordance with anatomical data, as well as outputs from layer 6, the feedback layer. We drive the model with drifting grating for visual stimulus, and we systematically located parameter regimes and modified network structure to match data and recreate a number of features exhibited by 4Cα. We studied orientation selectivity, firing rate statistics, modulation ratio, gamma band oscillation, and the relation of these properties with each other and with various aspects of network structure and parameters.

This paper is about neural mechanisms of direction selectivity (DS) in Macaque primary visual cortex, V1. DS arises in V1 layer 4Cα which receives afferent input from the Magnocellular division of the Lateral Geniculate Nucleus (LGN). LGN itself, however, is not direction-selective. To understand the mechanisms of DS, we built a new computational model (DSV1) of 4Cα. DSV1 is a realistic, large-scale mechanistic model that simulates many V1 properties: orientation selectivity, spatial and temporal tuning, contrast response, and DS. In the model, DS is initiated by the dynamic difference of OFF and ON Magnocellular cell activity that excites the model’s layer 4Cα the recurrent network has no intra-cortical direction-specific connections. In experiments – and in DSV1 -- most 4Cα Simple cells were highly direction-selective but few 4Cα Complex cells had high DS. Furthermore, the preferred directions of the model’s direction-selective Simple cells were invariant with spatial and temporal frequency, in this way emulating the experimental data. The distribution of DS across the model’s population of cells was very close to that found in experiments. Analyzing DSV1, we found that the dynamic interaction of feedforward and intra-cortical synaptic currents led to cortical enhancement of DS for a majority of cells. In view of the strong quantitative agreement between DS in data and in model simulations, the neural mechanisms of DS in DSV1 may be indicative of those in the real visual cortex. Motion perception is a vital part of our visual experience of the world. In monkeys, whose vision resembles that of humans, the neural computation of the direction of a moving target starts in the primary visual cortex, V1, in layer 4Cα that receives input from the eye through the Lateral Geniculate Nucleus (LGN). How Direction-Selectivity (DS) is generated in layer 4Cα is an outstanding unsolved problem in theoretical neuroscience. In this paper, we offer a solution based on plausible biological mechanisms: We present a new large-scale circuit model in which DS originates from slightly different LGN ON/OFF response time-courses and is enhanced in cortex without the need for direction-specific intra-cortical connections. The model’s DS is in quantitative agreement with experiments.