BioMed Research International

Volume 2016 (2016), Article ID 2769698, 12 pages

http://dx.doi.org/10.1155/2016/2769698

## Information in a Network of Neuronal Cells: Effect of Cell Density and Short-Term Depression

^{1}Department of Experimental and Clinical Medicine, University of Magna Graecia, 88100 Catanzaro, Italy^{2}King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia^{3}Department of Electrical Engineering and Information Technology, University of Naples, 80125 Naples, Italy

Received 17 December 2015; Accepted 10 May 2016

Academic Editor: Maria G. Knyazeva

Copyright © 2016 Valentina Onesto et al. 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

Neurons are specialized, electrically excitable cells which use electrical to chemical signals to transmit and elaborate information. Understanding how the cooperation of a great many of neurons in a grid may modify and perhaps improve the information quality, in contrast to few neurons in isolation, is critical for the rational design of cell-materials interfaces for applications in regenerative medicine, tissue engineering, and personalized lab-on-a-chips. In the present paper, we couple an integrate-and-fire model with information theory variables to analyse the extent of information in a network of nerve cells. We provide an estimate of the information in the network in bits as a function of cell density and short-term depression time. In the model, neurons are connected through a Delaunay triangulation of not-intersecting edges; in doing so, the number of connecting synapses per neuron is approximately constant to reproduce the early time of network development in planar neural cell cultures. In simulations where the number of nodes is varied, we observe an optimal value of cell density for which information in the grid is maximized. In simulations in which the posttransmission latency time is varied, we observe that information increases as the latency time decreases and, for specific configurations of the grid, it is largely enhanced in a resonance effect.

#### 1. Introduction

Networks of nerve cells are complex systems in which a large number of components combine to yield collective phenomena with improved abilities in contrast to simple components of that system [1–6]. The human brain itself is a grid or a network of bewildering complexity where 10^{12} neurons cluster in three-dimensional architectures. The unprecedented functions of human brain, including self-consciousness, language, and the development of memory, may depend less on the specialization of individual neurons and more on the fact that a large number of them interact in a complex network [2, 3, 5, 6]. The human brain and mechanisms of information propagation through neural nets are being heavily investigated in the last years. Emerging nanotechnologies, whereby surfaces with a controlled nanotopography can regulate and guide the organization of neuronal cells into complex networks [7–13], advancements in traditional disciplines, that is, computer science and information theory [6, 14–19], and the combination of the two [5], may provide scientists with new tools to elucidate the mechanisms through which the brain marshals its millions of individual nerve cells to produce behavior and how these cells are influenced by the environment.

The exchange of information between individual neurons is mediated by a cascade of chemical to electrical signals which travel across the gap (synaptic cleft, approximately 20 nm wide) between those neurons [20]. At similar synapses, an action potential generated near the cell body propagates down the axon where it opens voltage-operated Ca^{2+} channels. Ca^{2+} ions entering nerve terminals trigger the rapid release of vesicles containing neurotransmitter, which is ultimately detected by receptors on the postsynaptic cell [20]. The described process continues repeatedly until the response at the postsynaptic sites reaches and surpasses a limiting value (i.e., a threshold); then, the target neuron produces an impulse (an action potential) that propagates in turn to another neuron. Noticeably, information is encoded by the frequency of the action potentials generated by the neurons rather than by their intensity [21]. Individual neurons and electrical activity thereof are correctly described by the celebrated leaky integrate-and-fire model in which the membrane potential of a neuron obeys a function of the sole time [21–25]:where is the capacitance of , the membrane, is its conductance, and is the resting potential of the neuron. In (1), the current represents the stimulus that excites the neuron until the membrane potential reaches a threshold ; then, an action potential AP is generated and the system is maintained for a refractory time at rest, in which (1) does not hold anymore, and this accounts for the short-term synaptic depression of the neuron [22]. Notice that one can multiply both terms of (1) by the reciprocal of the conductance , which yields a different form of (1):in which is the time constant in a circuit theory interpretation of the neuron [21]. Equation (2) is used to predict the time evolution of individual neurons. Similarly, neurons in a grid are described by a set of coupled differential equations that generalize the model above to an ensemble of a large number of simple units connecting to each other. In contrast, in neural mass models, the activity of the entire neural population is lumped in a limited and generally low number of variables or parameters, in a statistical approach [21, 26]. These parameters are related to the moments of the distributions that are used to describe the neural population and may be sometimes coincident with the sole center of mass. While advantageous for mathematical convenience and computational tractability, neural mass models and their more sophisticated evolutions that have been developed over time (including mean field models and neural field models) are however based on an approximation and may therefore fail to resolve the dynamics of a system of neurons over each of its scales.

Here, we revise the integrate-and-fire model in the version proposed by De La Rocha and Parga [22] to extend the analysis to a bidimensional set of neurons in a grid. We use information theory variables, including the Shannon information entropy, described, for example, in [14, 16, 27, 28] and recapitulated in the following part of the paper, to determine the response of the network associated with an external stimulus. The intensity and distribution of information over the network are determined as a function of the total number of neurons in the grid (thus, cell density) and time of synapse recovery after the stimulus (thus, short-term depression). In what follows, neurons are connected through not-intersecting edges; in doing so, the degree of the graph would not depend on the number of nodes in the graph. Moreover, the maximum intermodal distance is upper bounded and maintained below a cut-off distance which represents an ideal synaptic length, whereby all connections greater than the prescribed cut-off are disrupted. In simulations where the number of nodes is varied over a significant range, we observe two different regimens of information dynamics in the grid: in the low cell density range, the information quality and density in a grid increase with ; in the high cell density range, the information content in the grid increases less rapidly than the total number of cells, meaning that the information quality and density decrease with . For intermediate cell densities (i.e., when all connections in the graph are realized) the information density and quality in the grid reach a maximum. In simulations in which the posttransmission latency time is varied, we observe that information increases as the latency time decreases. More important than this, we observe that when the average firing rate of individual neurons (i.e., a property of single neurons in isolation) is an integer number of times greater than the characteristic signalling frequency in the grid (i.e., a property of a set of neurons in cooperation), the transport of information is largely enhanced, similarly in concept to the resonance of a mechanical system. These data reinforce the view that the organization of neural cells in a network and the topology of the network itself play a major role in the spread of information in a complex of those cells.

#### 2. Methods

##### 2.1. Generating Networks of Neurons in the Plane

We consider neural cells uniformly distributed in a square domain with edge (Figure 1). Individual nodes are indicated with the symbol . In what follows, we may use interchangeably the terms neural cells, neurons, and nodes. Nodes are connected through not-intersecting lines or edges, which are the vertices of the Delaunay triangulation of those nodes in the plane (Figure 1). The number of edges resulting from a similar triangulation varies linearly with , with , where is the number of points on the convex hull of the original data set . Thus, the degree of the graph would not depend on the absolute number of cells and for sufficiently large , . The Delaunay triangulation is the dual graph of the Voronoi diagram. From this, certain properties arise, including the following: (i) the Delaunay triangulation of maximizes the minimum angle over all triangulations of ; (ii) the circumcircle of any triangle in the Delaunay triangulation is empty (contains no sites of ); (iii) the closest pair of sites in are neighbors in the Delaunay triangulation; (iv) the minimum spanning tree of is a subgraph of the Delaunay triangulation of . Recalling that the* Euclidean minimum spanning tree* is the graph with minimum summed edge length that connects all points in , the Delaunay triangulation of a set of neurons has therefore some biological sense. A system of neural cells on a substrate is likely to develop synapses following the shortest path between those cells, thus maintaining the energy of the ensemble at a minimum, which is equivalent to the conditions from (i) to (iv) above and especially to property (iv). For each element (node) in , we define a bundle of and indicate with the subset of nodes in that are connected to . The information about the connections amid the nodes in a graph is contained in the adjacency matrix , where the indices and run through the number of nodes in the graph. (where is the Euclidian distance function) if there exists a connection between and ; otherwise. In the analysis, reciprocity between nodes is assumed, and thus if information can flow from to ; it can reversely flow from to . In the framework of graph theory, we call a similar network an undirected graph. Notice that this property translates into symmetry of with . Moreover, .