# On Synchronizing Coupled Retinogeniculocortical Pathways: A Toy Model

Authors: B. L. Mayer; L. H. A. Monteiro
Journal: Computational Intelligence and Neuroscience
 (2018)

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## 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.


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## Body

## 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 60–120 Hz; for dynamic stimuli, synchronized oscillations in retina and LGN also occur at 60–120 Hz, but in VC at 30–60 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 Section2, 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 withN rows and N columns, in which the N2=n 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 2N(N-1) regular edges. Then, m extra edges are randomly included. Obviously, the higher the value of m, the lower the average shortest path length l. For instance, for N=10, then l=6.67 for m=0, l=5.81±0.17 for m=3, l=4.95±0.13 for m=9, l=3.99±0.10 for m=27, and l=3.14±0.03 for m=81. 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 N2 nodes, 2N(N-1) regular edges and m 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 bym′ 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 m′′ 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 P coming from retinal afferents are applied to either one or all N 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.Figure 1
Neural network built by takingN=10 and m=m′=m′′=9. The input is applied to neurons in the first row of the lateral geniculate nuclei. These neurons are denoted by black circles. An undirected edge is represented by double line; a directed edge, by single line.At the time stept, each node can be in one of four states: susceptible (S), active-1 (Y), active-2 (Z), or inhibited (R). The input of ith node at t is given by ∑jwijoj(t)+Ii(t), in which the index j labels the nodes linked to the ith node, wij is the synaptic weight connecting the jth to the ith node, oj(t) is the output of the jth node, and Ii(t) denotes the stimulus coming from the retina (which can be nonnull only for nodes in the first row of the nuclei). If the jth node is active (i.e., if it is in the state Y or Z) at t, then oj(t)=1; otherwise, oj(t)=0.The time evolution of the activity of this neural network is driven by the following rules of state transitions [9]. A S-node at t becomes a Y-node at t+1 if the input received at t is equal to or greater than the threshold T. A Y-node at t becomes a Z-node at t+1 if its input at t remains equal to or greater than T; otherwise, it returns to being a S-node. A Z-node at t becomes a R-node at t+1 independently of its input. A R-node at t becomes a S-node at t+1 and it is ready to fire at t+2, if the input received at t+1 is strong enough. Note that these rules are deterministic.Letx1(t), x2(t), x3(t), and x4(t) be the percentages of active nodes in the left-LGN, left-VC, right-LGN, and right-VC, respectively. Let pi(i=1,2,3,4) be the period of the oscillation found in xi(t) in permanent regime (i.e., after the transient phase), in a given time window (here, a time interval of 100 time steps). The whole network is considered to be synchronized if p1/r1=p2/r2=p3/r3=p4/r4, in which ri are integer numbers.We assume thatwij=1; therefore, all synapses have the same weight and they are excitatory. Inhibition, however, is a feature that can be necessary for biological neural systems achieving synchronous behavior [23–25]. In this model, inhibitory effects are taken into account in the transition Z→R. Due to this transition, a node firing at t and t+1 is inhibited at t+2; that is, it is forced to inactivity at t+2. The influence on the dynamics of the number d of consecutive time steps that a node can remain active was already explored [9]. It was found that the lower the value of d, the stronger the inhibitory effects; consequently, the number of simulations without synchronization increases with d. Here, d=2.The states of all nodes are simultaneously updated throughout a simulation. The results obtained in1.28×106 simulations are reported in the next section.


## 3. Simulation Results and Discussion

Consider that one time step of the simulation is equivalent to one millisecond of real time, because this is the time interval related to the state transitionsS→Y and R→S in actual neurons [10]. Consider also that a stimulus with unit magnitude and period P=10 ms is applied in the first row of the graphs representing the nuclei either to a single node or to all N nodes. The first kind of stimulus is denoted as k=1; the second one, as k=N. Observe that a forcing frequency of (1/10) ms-1 = 100 Hz is similar to those typically recorded in retina [3, 7, 8]. Computer simulations were performed by taking T=1, N=10 (thus, each one of the four graphs is composed by 100 nodes) and {m,m′,m′′}={3,9,27,81}. With these amounts of random links, the percentages of thalamocortical synapses and of callosal synapses related to cortical neurons are comparable to those found in actual nervous systems, as mentioned below.Fork=1 (i.e., stimulus applied to a single node in each pathway), for each triplet (m,m′,m′′), 100 random networks were created and all 100 combinations of possible inputs were simulated (100 combinations, because there are 10 nodes in the left hemisphere and 10 nodes in the right hemisphere that can receive the stimulus). Therefore, 6.4×105 simulations were accomplished. For k=10 (i.e., stimulus applied to 10 neurons in each pathway), for each triplet (m,m′,m′′), 10000 random networks were created, in order to execute the same number of simulations as for k=1. Thus, another lot of 6.4×105 simulations was run.Letη1k be the number of simulations in which P=p1=p2=p3=p4=10 and η2k is the number of simulations with P=p1=p3=10 and p2=p4=20 or 30, for k=1,10. Figure 2 shows an example of a simulation of type-η11 (retina, nuclei, and cortices oscillating at the same frequency 1/P, due to a periodic input of period P applied to a single neuron in each pathway), Figure 3, an example of type-η21 (retina and nuclei at the frequency 1/P, and cortices at a frequency that is half or a third of 1/P). Figure 4 exhibits an example in which synchronization did not occur; that is, at least one graph did not achieve a periodic behavior in the time window of 100 time steps (100 milliseconds of real time). Figure 5 illustrates the case η210.Figure 2
Time evolutions ofx1(t), x2(t), x3(t), and x4(t), which are the percentages of active nodes in the left-LGN, left-VC, right-LGN, and right-VC, respectively. Parameter values: T=1, N=10, k=1, P=10, m=m′=9, and m′′=27. Observe that, in the four random graphs, the percentages tend to a periodic oscillation with period equal to P.Figure 3
Time evolutions ofx1(t), x2(t), x3(t), and x4(t). Parameter values: T=1, N=10, k=1, P=10, m=9, and m′=m′′=27. In this case, p1=p3=10 and p2=p4=20.Figure 4
Time evolutions ofx1(t), x2(t), x3(t), and x4(t). Parameter values: T=1, N=10, k=1, P=10, m=3, and m′=m′′=81. In this case, p1=p2=30, p3=10, but x4(t) did not achieve a periodic behavior in the considered time window. Thus, the whole network did not synchronize.Figure 5
Time evolutions ofx1(t), x2(t), x3(t), and x4(t). Parameter values: T=1, N=10, k=10, P=10, m=m′=9, and m′′=27. In this case, p1=p3=10 and p2=p4=30.In order to evaluate the influence of the topological parametersm, m′, and m′′, we proceeded as follows. For each value of m, we added the numbers corresponding to η1k and η2k obtained for all combinations of m′ and m′′. For instance, for m=3ek=1, we added the numbers of η11 obtained in 160000 simulations with m′={3,9,27,81} and m′′={3,9,27,81}; then, we added the numbers corresponding to η21. A similar procedure was carried out for evaluating the influences of m′ and m′′ on the synchronization of the whole network.Table1 presents the results. The first obvious observation is that η110>η11 for any parameter values. Thus, input applied to k=10 neurons in each pathway enhances the number of cases in which the whole network is synchronized at the frequency 1/P, as compared to input applied to only k=1 neuron. The relation η210>η21 is also valid for most cases (it does not hold only for m=81). Thus, the higher the number of nodes in LGN receiving the input, the higher the number of synchronized networks.Table 1
Number of cases classified asη11, η21, η110, and η210 in function of m, m′, and m′′ for coupled visual pathways with T=1, N=10, and P=10. In each line, η1k+η2k<160000, because no synchronization and synchronization with period relationships distinct from those labeled as η1k and η2k were found.
η 1 1 η 2 1 η 1 10 η 2 10 m = 3 1206 311 130588 16947 9 21308 5211 136631 19991 27 112597 24083 133960 25606 81 139869 19728 141031 18905  m ′ = 3 78265 3417 151341 4898 9 62755 18681 112136 39408 27 58819 22241 129501 28236 81 75141 4994 149232 8907  m ′ ′ = 3 69249 11295 138688 17440 9 69843 12299 136990 18552 27 68512 12836 134758 21170 81 67376 12903 131774 24287About the influence ofm′′, which is the number of callosal links connecting the cortices: η1k tends to decrease and η2k tends to increase with m′′. This effect is more pronounced for k=10 than for k=1. Therefore, VC synchronously oscillating at a lower frequency than LGN can be favored by increasing both m′′ and the number of nodes in LGN receiving the input.About the influence ofm and m′, which are, respectively, the amount of random links inside each graph and the amount of thalamocortical links: the dependencies of η1k and η2k on m and m′ are not monotonous. Note that the maximum numbers of η1k are found for extreme values of m and m′: for k=1 and k=10, the maxima occur for m=81 and m′=3. The maximum numbers of η2k occur for intermediate values of m and m′: for k=1, the maxima are found in m=27 and m′=27; for k=10, in m=27 and m′=9.TakeN=10 and m=m′=m′′=27. Thus, there are 4N(N-1)+2m+2m′+m′′=495 connections involving cortical nodes. In this network, the percentage of thalamocortical synapses is 54/495≃11%, and the percentage of callosal synapses is 27/495≃5%. Interestingly, similar values can be found in literature. For instance, in mouse cortex, thalamocortical synapses are about 15% [26]; callosal synapses, about 5% [27]. One can conjecture that the values of m, m′, and m′′ found in actual visual pathways favor the occurrence of synchronization between LGN and VC, in order that the frequency observed in VC is equal, half, or a third of the one measured in LGN.In our toy model, for obtaining synchronization between LGN and VC in distinct frequencies, the random links connecting them must be directed. If undirected links are used to connect LGN and VC, simulations show that either the four graphs synchronize with period equal toP or they do not synchronize.


## 4. Conclusions

Toy models can be valuable, because they can reveal which features of the studied system are most relevant. Here, a toy model was employed to simplistically investigate the dynamics of coupled visual pathways. Only three neural structures were taken into consideration: retina (represented as a periodic input), LGN, and VC (both represented by Newman-Watts graphs). The aim was to determine how the wiring topologies affect their synchronous activities, because synchronization among these neural structures is supposed to be underlying the visual perception [1–8].Callosal connections can be responsible for integrating interhemispheric information, binding perceptual features and enabling a coherent representation of a visual scene [28]. We found that η2k significantly increases with m′′ when the stimulus is applied to all nodes in the first row of both pathways; if it is a applied to a single node, then m′′ barely affects the synchronization. Thus, interhemispheric synchronization can be improved not only by increasing m′′, but also by increasing the number of stimulated nodes. It is interesting to note that η1k tends to decrease with m′′. Thus, increasing the amount of callosal connections favors that the cortices synchronously oscillate in a frequency lower than the one detected in the nuclei.A key component of the neural coding is the timing of spiking activity [29]. This timing is influenced by the neuron features and by the synaptic connectivity [30, 31]. In this context, our model shows that there are values of m and m′ favoring the cases labeled as η1k and η2k. For instance, according to Table 1, the maxima of η21 and η210 occur for m=27. By taking m=m′=m′′=27, then 11% of synapses involving cortical neurons are thalamocortical ones and 5% are callosal ones, which are values compatible with those found in actual nervous systems [26, 27]. Notice that the validity of our results relating the amounts of synaptic connections (the parameters m, m′, and m′′) to synchronous behaviors classified as η1k and η2k can be experimentally tested.



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