Computational Intelligence and Neuroscience

Volume 2018, Article ID 6858176, 6 pages

https://doi.org/10.1155/2018/6858176

## On Synchronizing Coupled Retinogeniculocortical Pathways: A Toy Model

Correspondence should be addressed to L. H. A. Monteiro; rb.eiznekcam@mziul

Received 31 August 2017; Revised 28 January 2018; Accepted 6 February 2018; Published 8 March 2018

Academic Editor: Christian W. Dawson

Copyright © 2018 B. L. Mayer and L. H. A. Monteiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A Newman-Watts graph is formed by including random links in a regular lattice. Here, the emergence of synchronization in coupled Newman-Watts graphs is studied. The whole neural network is considered as a toy model of mammalian visual pathways. It is composed by four coupled graphs, in which a coupled pair represents the lateral geniculate nucleus and the visual cortex of a cerebral hemisphere. The hemispheres communicate with each other through a coupling between the graphs representing the visual cortices. This coupling makes the role of the corpus callosum. The state transition of neurons, supposed to be the nodes of the graphs, occurs in discrete time and it follows a set of deterministic rules. From periodic stimuli coming from the retina, the neuronal activity of the whole network is numerically computed. The goal is to find out how the values of the parameters related to the network topology affect the synchronization among the four graphs.

#### 1. Introduction

Unveiling how nervous systems perform cognitive and sensory functions depends on understanding how stimuli from the outside world are translated into neuronal responses. For instance, in mammalian visual system, synchronized responses underlie image perception [1–8]. In fact, in mammals, visual stimulations evoke synchronous neuronal activities in retina, lateral geniculate nucleus (LGN) of the thalamus, and visual cortex (VC), at a time scale of tens of milliseconds [1–8]. For static stimuli, typical synchronization frequencies detected in these three structures are Hz; for dynamic stimuli, synchronized oscillations in retina and LGN also occur at Hz, but in VC at Hz [3, 7, 8]. Therefore, the cortical frequency band is usually equal to or lower than the retinal/thalamic frequency band.

Frequency transitions in a single retinogeniculocortical pathway were already numerically investigated [9]. Here, we consider that the visual system of mammals is indeed composed by two pathways coupled by the corpus callosum, which connects the cerebral hemispheres. The aim is to examine how the amount of links connecting these neuronal structures influences the emergence of synchronized responses in LGN and VC, with equal or distinct frequencies.

As the brain has a modular architecture [10], there are many theoretical studies on neuroscience based on coupled neural oscillators [11–14]. In this work, Newman-Watts random graphs [15] are used to represent the network topology of the oscillators composing the visual pathways; and the discrete time evolution of the states of these oscillators is governed by deterministic rules, as in other models on neurodynamics [16–19].

This manuscript about synchronization of oscillatory neuronal responses is organized as follows. In Section 2, the model is described. In Section 3, results obtained from numerical simulations are presented and discussed. In Section 4, the conclusions are stressed.

#### 2. The Model

The whole network of our toy model is created as follows. First, consider a square lattice with rows and columns, in which the nodes are linked in a cross-like coupling pattern. Thus, each node has four regular neighbors (except the nodes placed at the boundaries, which have only two or three neighbors) and there are regular edges. Then, extra edges are randomly included. Obviously, the higher the value of , the lower the average shortest path length . For instance, for , then for , for , for , for , and for . Since this Newman-Watts-type graph has small-world features [15, 20], it can be suitable to model biological neural structures [21, 22]. Hence, this undirected graph with nodes, regular edges and random edges, is used to represent the LGN. It is also used to represent the VC.

In a hemisphere, the LGN is coupled to the VC by random edges directed from the LGN to the VC. The dynamics of this single visual pathway were already examined [9]. In this work, the hemispheres are coupled by undirected random edges connecting the cortices. Thus, there are two retinogeniculocortical pathways coupled by callosal connections.

Static and dynamic visual stimuli are encoded as coherent oscillations by the ganglion retinal cells, which are the output neurons of retina [10]. Hence, periodic stimuli with period coming from retinal afferents are applied to either one or all nodes in the first row of the graphs representing the nuclei, as illustrated by Figure 1. Then, the activity of the whole network is determined at each time step.