Dynamics of large cooperative pulsed-coupled networks

We study the deterministic dynamics of networks N composed by m non identical, mutually pulse-coupled cells. We assume weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of m. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural"spiking period"\ p>=1, and that if the cells are mutually structurally identical or similar, then the synchronization is complete (p= 1) . Second, we prove that the amount of information H that N generates or processes equals log p. Therefore, if N completely synchronizes, the information is null. Finally, we prove that N protects the cells from their risk of death.


1.
Introduction. The theory of deterministic dynamical systems composed by two or more coupled dynamical units assumes that each unit -which we call cell -has a proper own dynamics, and that the couplings among the units are interactions that depend on the instantaneous states of the cells [33,7].
Among the systems of interacting units, we focus on those that are pulsed-coupled (i.e. the interactions are instantaneous). In particular, the global system can be understood as a multi-dimensional differential or difference equation with impulsive terms [18].
On the one hand, the theory of dynamically interacting units is a source of mathematical open questions [33]. In particular, those systems governed by impulsive differential equations [28] pose mathematical problems that are mostly open, except in particular cases or low dimensions.
On the other hand, the network of interacting dynamical units is a particular model of a coalitional game, that evolves or changes on time. We will discuss this relation in Subsection 1.2.
The dynamical systems composed by mutually interacting units, and in particular those that are pulsed-coupled, have relevance in many applications. As examples in Physics, a two-dimensional impulsive differential equation models the joint dynamics of two or more coupled oscillators [27]. In particular, they are used in applications to Light-Controlled-Oscillators (LCO), [25]. The mathematical investigation of a genetic regulatory network of two antagonist genes, is also modeled as a network of pulse-coupled units [9]. In Neuroscience, among the theoretical methods of research, the mathematical analysis of the dynamics of pulse-coupled networks is applied [12,16]. In Engineering, networks of coupled dynamical units are designed for control systems and communications [32]. Computational research on artificial intelligence, by means of artificial neuronal networks, is used to analyze, simulate, and investigate on data obtained from dynamical systems of interacting units with a large degree of complexity [8]. In Economics, networks of coupled units are used to investigate the equilibrium states in social systems of interacting agents [1]. Artificial neuronal networks are applied for the prediction of the exchange market [21], also to investigate on financial markets [17], and for the accounting of financial applications [15]. In Ecology, networks of interacting units model the dynamics of predator-prey communities of two or more species [30,13]. The dynamics of an infectious disease, taking into account the interaction among populations of diverse infectious agents, is modelled as a neural network [26]. In Geosciences, the forecast of ozone peaks in weather prediction uses computational methods on artificial neuronal networks [10]. In Social Sciences, the dynamics of large WWW social networks is mathematically modelled by the interactions of their individuals [22,31]; and the self-synchronization of many small clusters of cells in a low-dimensional network models the dynamics of a medieval social network [3].
1.1. The object and method of research. Along this paper we investigate, by exact mathematical analysis and deductive proofs, the global dynamics of certain pulse-coupled deterministic networks of m cooperative cells, for large values of m.
Each cell i ∈ {1, 2, . . . , m} is governed by a deterministic dynamical sub-system, which -if i were hypothetically isolated from the network -we call the free dynamics of i. Besides, each cell i acts on the other cells j = i of the network at certain instants t i , which we call the spiking or milestone instants of i. Conversely, each cell i receives the actions from the other cells j = i at the spiking instants t j of j.
The free dynamics of each cell evolves governed by a finite-dimensional ordinary differential equation, joint with an autonomous and instantaneous reset or update rule. The update rule applies when the state of i arrives to a pursued goal or threshold level θ i . The update rule resets the state of i, or equivalently, it changes the velocity according to which the free dynamics evolves. The free dynamics of the many cells of the network may be mutually different. As a particular case, in Neuroscience the model of the free dynamics of each cell -by integration of a differential equation plus a reset or update rule -is called integrate and fire. Each reset or update event of the cell (the neuron) is called a spike. In brief, the spike of a cell i is produced when its state arrives to the goal θ i .
The cells compose the network by mutual interactions between any ordered pair (i, j) such that 1 ≤ i, j ≤ m and i = j. These interactions exist in both directions (some interactions may be zero), but are neither necessarily symmetric nor simultaneous. Roughly speaking, the action from the cell i to the cell j = i is a discontinuity jump ∆ i,j -applied on the state of j -that is produced at the instant t i when i spikes. So, the instant t i depends on the state of i. The rule is the same to define the action from the cell j to the cell i, but the matrix (∆) i,j is not necessarily symmetric. Besides, the instant t i when the action ∆ i,j is applied, is in general different from the instant t j when the action ∆ j,i is applied.
Our purpose of research is to find qualitative and quantitative relevant characteristics of the global dynamics of such an abstract network, while time t evolves to the future. As said above, the methodology is to find the exact abstract mathematical statements and their deductive proofs.
We take the main ideas from [4], in which a model of a network, composed by integrate and fire biological neurons, is studied by the exact mathematical method.
The main differences between the dynamical system that we study here and the one studied in [4], are the following: First, we assume that the cells are cooperative (which in Neuroscience are called excitatory). This means that ∆ i,j ≥ 0 for all ordered pairs (i, j) such that i = j, and the value zero is admitted in some of our results. In [4], any sign of ∆ i,j is admitted by hypothesis, but only nonzero values are assumed. Second, we do not assume that the free dynamics is the same for all the cells. In [4] all the cells are identical. Third, we neither assume the linearity of the differential equation that governs the free dynamics of each cell, nor the existence of a Lyapunov stable equilibrium state for the solution flow of this differential equation. In [4] these latter two conditions are assumed.
1.2. The network as a cooperative game that evolves on time. The mathematical model of the network N that we study in this paper is also an evolutive game represented by a graph whose m vertices are the players i ∈ {1, . . . , m} (the cells), and whose edges ∆ i,j (the interactions) are directed and weighted. The hypothesis of cooperativity among the individuals, namely ∆ i,j ≥ 0 for all i = j makes the game work in an imitate the best strategy, which produce players that adopt a myopic behaviour ( [14]). In fact, by hypothesis, each cell or player i just knows its own actions to the other players j = i, the value of its own satisfaction variable, and the actions it receives from the other cells. But i ignores the global state and dynamical behaviour of the whole network.
The model of the network N , as described in Subsection 1.1 from the dynamical viewpoint, is a cooperative or coalitional game, that changes on time. It disregards the individual strategies of its cells (or players) and instead, it focusses on the coalitions (which we call clusters of cells), defined as nonempty subsets of cells i ∈ {1, . . . , m} whose satisfaction variables S i arrive simultaneously to their respective goals or threshold levels θ i > 0.
The characteristic function ν : 2 m → R + of the coalitional game (i.e. the function assigning the total gain or payment ν(A) to each coalition A ⊂ {1, . . . , m}, with the agreement ν(∅) = 0) can be defined as the sum of the goal levels θ i of the cells i ∈ A. In fact, at each instant t n for which a nonempty coalition I n is formed (i.e. a cluster of simultaneously spiking cells is exhibited), the satisfaction variable of each cells i ∈ I n equals its respective goal or threshold level θ i . Since θ i > 0 for all i ∈ N , the coalitional game is convex, i.e.
We note that, since in our case ν(A) = i∈A θ i , inequality (1) is indeed an equality. Thus: In other words, the gain for a player i for joining a coalition B larger than A, is non negative.

ELEONORA CATSIGERAS
As for any convex coalitional game, N has a nonempty core of solutions. In Game Theory, a solution in the core is a vector (z 1 , z 2 , . . . , z i , . . . , z m ) ∈ R m , which is called "allocation" or "payoff vector", such that i∈I For our model, the payoff vector at the n-th. instant t n when at least one cell arrives to its goal level and spikes, can be defined by the following formula: is the value of the satisfaction variable S i of the cell i just before instant t n (cf. Definition 2.1), I n is the cluster at the spiking instant t n (cf. Definition 2.7) and ∆ j,i ≥ 0 is the cooperative action from the cell j ∈ I n to i = j at any instant for which j spikes. In Equation (10) we will precisely state the rule according to which the cooperative interactions ∆ j,i among the cells of the network increase their satisfaction variables. In brief, the cell i spikes at instant t n , hence it belongs to the cluster I n , if and only if one of the following conditions is satisfied: either the satisfaction variable S i arrives to the goal value θ i spontaneously (due to the free dynamics of i) at instant t − n , or the satisfaction variable at instant t − n is smaller than θ i but suddenly increases to become larger or equal than θ i , by the adding of some positive cooperative actions ∆ j,i from cells j = i that spike at the same instant t n . In any case, the component z i of the payoff vector is z i ≥ θ i if z i ∈ I n . Since ν(I n ) = i∈In θ i we deduce that Inequality (2) holds in the coalitional game N .
Although the characteristic function ν is assumed to be invariant with time t, the payoff vector changes with t, because the satisfaction variables evolve on time, and the interactions among the cells are not constantly applied at any time. In fact ∆ j,i is effectively added to the satisfaction variable S i just at the instants for which the cell j spikes. So N is an evolutive coalitional game: its solutions depend on time.
In Definition 3.1 we define the global synchronization phenomenon of the network N as the recurrent exhibition -infinitely many times in the future -of the so called "grand coalition". Precisely, all the cells of the network compose a single cluster: all of them spike simultaneously, and this phenomenon occurs infinitely many times. In Theorem 3.5 we prove that, if the number m of players is large enough in relation with the minimum positive interaction (namely, m is large compared with the minimum payoff component at any instant), then the global synchronization occurs recurrently. Nevertheless, between two instants t * 2 > t * 1 when the grand coalition is exhibited, there may appear many coalitions or clusters that are smaller than the grand coalition. In Theorem 3.20 we prove that if more different coalitions recurrently appear in the future, then the amount of information of the global dynamics of the network increases.
In spite that the network can be studied as a coalitional evolutive game, it has some important differences from the models that are usually investigated in Game Theory, even from those that focus on evolutive games (e.g. [14,1,19]): For instance, in our model, each cell or player i decides the instants when it acts in the network, by integrating the states of its own internal dynamics with the actions (or payoffs) that it had received from the other players. So, there does not exist a forcing external mechanism making the players interact sharing a resource. As a counterpart, neither the cell nor the coalition, has an alternative to decide how to choose among several strategies, its actions on the network. The cells and the coalitions are not optimizers. The network is just a deterministic dynamical system whose state suffers instantaneous changes according to the states of its cells and, as a result of the positive interactions among the cells, it shows patterns of coalitions that spike simultaneously infinitely many times in the future.
1.3. The hypothesis. We adopt three types of hypothesis. First, we assume that the number of cells -in a part or the whole network -is large enough with respect to certain other parameters of the network (Inequality (14) of Theorem 3.5 and a similar inequality in Corollary 3.24). Second, we study the cases for which the cells are cooperative (cf. Definition 2.2). Third, we assume that the network's graph is complete, or, if not, that there exists a large cooperative core (i.e., a complete subgraph of cells which are fully cooperative, cf. Definition 3.23).
1.4. The results to be proved. Under the hypothesis above described, we prove three results: the periodic eventual synchronization, the positiveness of the protection factor of the network against the risk of death of its cells, and the value of the total amount of information that the network is able to generate or process.

1.4.1.
Results about synchronization. In Theorems 3.5, 3.6 and Corollary 3.24, we prove that the network necessarily self-synchronizes the spikes of clusters of cells periodically, after a transitory waiting time, and that this behaviour is robust (i.e. it persists under small perturbation of the numerical and functional parameters of the network). After the transients, the synchronized clusters are independent of the initial state. But the finite transitory time-interval itself, does depend on the initial state.
We give the word "synchronization" an exact mathematical meaning (cf. Definitions 3.2 and 3.1). By eventual periodic spike-synchronization, we mean that the interactions among all the cells within the same cluster are produced simultaneously, infinitely many times in the future and periodically -but not necessarily at all the spiking instants of the network (Definition 3.2). Nevertheless, this definition is rather non-standard in Physics. (For the concept of synchronization and phase locking in Physics see for instance [23].) In fact, the classical definition of synchronization requires all the cells be identical (e.g. [20]). But along this paper, we are not assuming the hypothesis of identical cells, neither in the whole network nor in a part of it.
Joining the results of Theorems 3.5 and Theorem 3.22, we prove that the events' synchronization of more than one cluster is produced only if the cells are mutually different. In other words, if all the cells are structurally identical -also if they are non identical but similar -we prove that all the events of the whole network synchronize, after a transitory time-interval and from any initial state (Theorem 3.22, Part (a)). In brief, networks of similar cooperative cells form a unique spikesynchronized cluster.
In their seminal article [20], Mirollo and Strogatz proved that networks of identical fully cooperative cells, with constant interaction ∆ i,j = ∆ > 0 ∀ i = j, synchronize all their spikes. In [2], Bottani proved a very general result under a certain stability hypothesis, stating the synchronization of complete networks composed by a large number of excitatory neurons (i.e. cooperative cells) that are modelled as integrate and fire oscillators: Either all the oscillators evolve synchronized in block, or subsets of synchronized oscillators appear always in stable avalanches. The synchronization of several clusters of cells, provide repetitive patterns that allow the network to organize the information. In Theorem 3.5 we generalize Bottani's results to mathematically abstract networks of pulse-coupled cooperative cells, that are neither necessarily identical nor similar, and without the stability hypothesis. In Corollary 3.24 we generalize the statements for some kind of not completely connected networks.

1.4.2.
Results about the protection and the amount of information. In Theorem 3.15 we prove that cooperative networks with a large number of cells protect their cells from external negative interferences, diminishing their risk of death (cf. [11]). This protection is due to the synchronization of sufficiently large clusters. As a counterpart, in Part (c) of Theorem 3.22, we prove that if a complete synchronization is achieved, then the total amount of information that the network can generate, or process, is null. But if the synchronization is not complete -as said above this occurs if perceptible differences exist among the free dynamics of the cells -then the network is able to process a positive amount of information (see Theorem 3.20, in the case that the spiking period p is not 1). Nevertheless, this amount of information is necessarily much lower than the theoretical maximum that a general network could process, according to the number of cells that are employed in the task. This is because, due to Theorem 3.5, in large cooperative networks there exist large clusters of cells whose spikes are produced all together. Therefore, they contribute to enlarge the number of different spiking-code patterns, as if each cluster had a unique cell.
The null amount of information of the cooperative networks that completely synchronize, opposes to many networks composed by all antagonist cells. These latter may exhibit a large amount of (rather unpredictable) information. See for instance the "virtual chaos" in [5,6], or the "stable chaos" in [24]. As a counterpart, the networks composed by all antagonist cells unprotect their cells and so, they enlarge their risk of death (cf. Remark 3.13). As a consequence, if all the interactions were antagonist, the whole network would be in risk, since the death of many of its cells can drastically diminish the initial richness of its global dynamics.
Therefore, a large network that performs both features: protects their cells and processes a large amount of not always structured and predictable information, should theoretically be composed by some cooperative cells and by some antagonist ones, in a more or less balanced interplay among them. Nevertheless, networks of antagonist cells (which in Neuroscience are called inhibitory neurons) also may synchronize and exhibit null amount of information, if the hypothesis of instantaneous spikes is released to take into account the delays [29].
Large networks of all cooperative cells with instantaneous interactions-the ones that we study along this paper -also show a balanced interplay between the amount of information and the protection of the network against the risk of death of its cells. In fact, in Theorem 3.22, we prove that the network maximizes its protection to the cells, if they have mutually similar free dynamics. But if so, the total amount of information becomes zero (Part (c) of Theorem 3.22). In other words, if the network is able to process a positive amount of information, then the cells are mutually different. In particular, the time-constants of their respective free dynamics must be diverse. In such a case, the network may show many different clusters of mutually synchronized cells, and a long period p until the spike patterns of the whole network repeat. In Theorem 3.20, we prove that the amount of information H that a fully cooperative network is able to generate or process, equals log 2 p. So, H is larger if the number of synchronized clusters and the period p are larger.
The paper is organized as follows: In Section 2 we pose the previous definitions and hypothesis that are assumed for the mathematical model of a cooperative network of pulsed coupled dynamical units. In Section 3, we include the particular mathematical definitions and the statements of the theorems that we will prove along the paper. From Section 4 to Section 8 we write the proofs.

Previous definitions and assumptions.
We call the scale of the cell the micro-scale relative to the whole network N under study. The state x i of a cell i isby hypothesis -governed by a continuous, deterministic and autonomous dynamical system Φ i , on time t ∈ R, t ≥ 0, which we call the free dynamics of the cell i. It evolves on a finite-dimensional, differentiable, riemannian and compact manifold X i . Therefore, x i is a (maybe multidimensional) variable living on a compact metric space with a local euclidean structure, such that for each instant t ≥ 0 and for each initial state x i (0), the continuous mapping Φ i satisfies the following equalities: The flow defined on a finite dimensional manifold by an autonomous ordinary differential equation is an example of a continuous dynamical system, and may model each cell. Each cell i has its own free dynamics, governed by its own rules. The spaces X i where the respective states x i of the cells live, and their dimensions, may be mutually different.
We call the scale of the whole network the macro-scale or also, the global scale. It is defined by the interactions ∆ i,j among any ordered pair (i, j), i = j of different cells. These interactions are produced, in our case, according to certain deterministic rules that we will state in Definition 2.1.
Each network N may be a cell of a larger hyper-macro network. We will not study this hyper-macro system, nor the role of N as a cell of it. Nevertheless, we assume that this hyper-macro system exists and may perturb the state x i of each individual cell i in N .
Two main concepts are the spikes or milestones and the eventual death of a cell.
. We call S i the satisfaction variable of the cell i. This variable may be, for instance, the algebraic sum of several positive or negative components that depend on the state x i .
By hypothesis, the satisfaction variable S i , while not perturbed from the exterior of the cell i, satisfies the following two conditions, (4) and (6): where

ELEONORA CATSIGERAS
We call θ i the goal level or threshold level. It may be achieved at many different states x i of the cell, i.e. the set S −1 i (θ i ) ⊂ X i is not necessarily reduced to a single point. Nevertheless, we assume, by hypothesis: By hypothesis, at instant 0 the initial state x i (0) of each cell satisfies: On the one hand, at each instant t i such that lim t→t − i S i (x i (t)) = θ i , the cell i reacts in the following way: First, the following reset or update rule holds: i.e. the state x i of the cell i jumps to x i,reset at intant t i . Thus, the satisfaction variable resets to zero at each milestone-instant. Second, the cell i "produces" a spike at each milestone-instant t i , i.e. i plays in the game N when its satisfaction variable S i arrives to the goal θ i . At instant t i the cell i sends instantaneous signals ∆ i,j to the other cells j = i.
On the other hand, when one and only one cell In such a case, by hypothesis, the state x i suffers a discontinuity jump such that: ) Finally, by hypothesis of the model, if all the cells of a nonempty set I n = {j 1 , . . . , j k } -of k different cells -spike simultaneously at some instant, say t n , and if ∆ j,i ≥ 0 for all j ∈ I n and for all i ∈ I n , then the state x i of any other cell i ∈ I n suffers a discontinuity jump such that:

Definition 2.2. (Cooperative and antagonist cells)
We call a network (fully) cooperative if all its cells are (fully) cooperative.
Formulae (9) and (10) show that when each cooperative cell j spikes, then it contributes to enlarge the values of the satisfaction variables S i of the other cells i = j. Thus, j helps the other cells i to approach to their respective goal levels, and so, it shortens the waiting times until the milestones of the others occur.
When an antagonist cell j spikes, it reduces the values of the satisfaction variables S i of the other cells Thus, j enlarges the waiting times of the others to arrive to their respective goal levels.
In this paper, we will focus on full cooperative networks or subnetworks. By conditions (4) and (5), while the free dynamics of a cell is not negatively perturbed by external agents, and while the satisfaction variable S i of the cell i does not arrive to its goal level, it strictly increases on time. In other words, each cell is "born optimist": if free, it approximates with positive velocity to its goal. Besides, the instantaneous velocity g i (x i ) is bounded away from zero, because min xi∈Xi g i (x i ) > 0 exists, due to the compactness of the space X i and the continuity of g i . So, if free, each cell arrives to its goal after a finite time. Nevertheless, if a negative term −δ i < 0 is added to the real function g i , the velocity dS i /dt will decrease, and may also become negative.

Definition 2.3. (Negative external interferences)
We call a negative number −δ < 0 negative differential interference to the cell i from its external environment, if during an interval of time, the differential equation (4) is substituted by We call a negative number −∆ < 0 negative impulsive interference to the cell i from its external environment, if at some instant t for which the satisfaction variable S i has not arrived to its goal level, the state x i of the cell suffers a discontinuity jump such that

Definition 2.4. (Death of a cell)
The death of a cell i (cf. [11]) occurs at an instant T ≥ 0, if for any time t > T the cell i does not spike. In other words, after a cell i dies, it does not arrive to its goal anymore, and so, it stops sending actions to the other cells of the network forever.
In Definition 3.7, we will state a numerical formula to measure the theoretical intrinsic risk R i of death of any cell under eventual negative interferences, if it were not connected to the cooperative network. In Definitions 3.11 and 3.12, we pose formulae to measure the net risk of death R i < R i , if the cell i is interacting in a cooperative network, and thus, the protection factor P i > 0 that the network provides to i.

Definition 2.5. (Space of parameters)
Let N be a network with m ≥ 2 fixed cells. The parameters of the network are: where, according to Definition 2.1, g i is the given C 1 real function in the second member of the differential equation (4) which governs the free dynamics of the cell i; θ i ∈ R + is its goal level; and ∆ i,j ∈ R is the interaction in the network from the cell i to the cell j = i.
In the space of all the parameters of a network with exactly m cells (for m ≥ 2 fixed), we define the topology induced by the following metric (distance), where the parameters of the networks N and N are defined by Equality (12); those denoted without (with ) correspond to the network N (resp. N ): In the above equality, · C 1 is the C 1 norm in the space of all the C 1 real functions defined on the compact manifold X i . We say that a phenomenon of the global dynamics is robust, or persistent, or structurally stable if the set G of parameters of the networks for which the phenomenon occurs is open. In other words, if a phenomenon is robust and if Param(N ) ∈ G, then Param(N ) still belongs to G for any sufficiently small perturbation N of the network N (openness condition).

Definition 2.6. (Spiking instants and interspike intervals of the network)
We denote by {t n } n∈N + the strictly sequence of all the instants for which at least one neuron spikes. We call t n the n-th. spiking instant of the network. We call (t n , t n+1 ] the n-th. interspike interval of the network and denote it by ISI (n) .

Definition 2.7. (Clusters or spiking codes)
We denote by I n the set of neurons that spike at the instant t n . We call I n the n-th. cluster or also, the n-th. spiking code.

Definition 3.1. (Spike-synchronization)
We say that the network eventually synchronizes spikes if there exists a subsequence {t n h } h∈N of spiking instants such that the respective clusters I n h are {1, . . . , m} (i.e. all the cells spike at instants t n h ).

Definition 3.2. (Periodic spike-synchronization)
We say that all the cells of the network eventually periodically synchronize spikes with period p ≥ 1, if there exists n 0 ≥ 0 such that: i) the subsequence {t n h } h∈N of Definition 3.1 satisfies ii) the sequence {I n } n≥0 of clusters satisfies iii) the sequence {t n+1 − t n } n≥0 of the interspike intervals' lengths satisfies: We call p the natural spiking period of the network.
Note that the network eventually periodically synchronizes spikes, if and only if at least one cell spikes at instant t n for all n ≥ 0, no cell spikes at instants t ∈ (t n , t n+1 ) for all n ≥ 0, and all the cells spike at instants t n0+hp for any natural number h ≥ 0.

Definition 3.3. (Transitory time)
For each fixed initial state for which the network periodically synchronize spikes, we call t n0 the waiting time or transitory time until the synchronization of the full network occurs.
Note that the occurrence of synchronization and the value of the waiting time depend on the initial state of the network.
Let N be a network composed by m ≥ 2 fully cooperative cells (cf. Definition 2.2). Recall that θ j denotes the goal level of the cell j, and ∆ i,j denotes the action from the cell i to the cell j (cf. Definition 2.1).

Definition 3.4. (Large cooperative network)
A fully cooperative network is called large if  Under the hypothesis of Theorem 3.5, the transitory time T and the natural spiking period p satisfy the following inequalities: We prove Theorem 3.6 in Subsection 4.2. Now, we define the risk of death of each cell and the protection factor of the network.
In Definition 2.4 we say that a cell i dies if (due to external causes of the cell) there is an instant T such that for all t > T the satisfaction variable S i remains smaller than the goal level θ i . In brief: Due to the rules of the interactions among the cells, if a cell i dies, it can not act on the network anymore (after time T ).
We measure the risk of death of the cell i by the following definitions: The intrinsic risk of death R i of the cell i, relatively to the other cells of the network is defined by where m is the number of cells of the network. Since θ i > 0 for all i, Equality (17) immediately implies the following: Recall Definition 3.16: the n-th. spiking-code I n is the (nonempty) set of cells that spike at instant t n . Let us fix a cell i. i } h≥1 of spiking times of the cell i is defined by the following equalities: We denote t  We call the time-interval When a cell i receives cooperative actions ∆ j,i > 0 from the other cells j = i of the network, then its satisfaction variable S i is increased, to approach (or even reach) its goal θ i . Equivalently, a positive action ∆ j,i > 0 can be understood, as a reduction of the goal level θ i substituting it by θ i − ∆ i,j . Roughly speaking, when a cell i receives positive interactions from the other cells, its risk of death diminishes.
We call ∆ (h) i,N the net action that the cell i receives during its h-th. inter-spikeinterval from the other cells of the network.
If the network is not cooperative, then some of the actions ∆ j,i in Formula (19) may be negative, so the sum of all of them may be negative or null. Note that, in general, the net action ∆ In general the net risk of death R i,N depends on the initial condition. For simplicity in the notation, in the sequel we will write R i to denote the net risk of the cell i, when the network N is clear from the context, and when referring to some fixed h ≥ 0.

Definition 3.12. (Protection factor P i )
The protection factor P (h) i,N of the network N to its cell i, during the h-th. interspike-interval of i, is In the sequel, we will simply write P i instead of P  that the cell i receives from the other cells of the network is positive (resp. negative). Therefore, if the network is fully cooperative, its protection factor is strictly positive, and if the network is fully antagonist, its protection factor is strictly negative.
In the following Proposition 3.14 we state that net risk R i of a cell that is connected to the network N , is (essentially) the product of its intrinsic risk R i , if i were not connected to the network, by 1 − P i .
In other words, if the protection factor P i of the network were positive, then the net risk R i is smaller than the intrinsic risk of death, and if the protection factor P i were 1, then the net risk of death R i would be zero. So, a cooperative network always reduces the intrinsic risk, and may also make it zero, if the cooperative interactions are large and frequent enough.
We prove Proposition 3.14 in Subsection 6.1.

Theorem 3.15. (Protection factor)
Under the hypothesis of Theorem 3.5, the protection factor P i of the network to each cell i is positive. So, the net risk of death R i of each cell i under negative external interferences, when it is connected to the network, is strictly smaller than its intrinsic risk R i when it is isolated from the network.
We prove Theorem 3.15 in Subsection 6.2.

Definition 3.16. (Code-patterns)
For any natural number k ≥ 1, and for any fixed initial state of the network, we construct the following word of clusters with length k: Π n,k := (I n , . . . , I n+k−1 ).
We call Π n,k the n−th. code-pattern with length k from the initial state x(0).

Definition 3.17. (Recurrent code-patterns)
We say that a code-pattern Π k with length k is recurrent, if there exists n j → +∞ such that Π nj ,k = Π k ∀ j ∈ N. We denote by P k the set of all the recurrent code-patterns with length k, obtained from all the initial states of the network. We denote by #P k ≥ 1 the cardinality of the set P k .

Definition 3.18. (Amount of information)
We denote where log 2 is the logarithm in base 2.
We call H the total amount of information that the network N is able to process.
Note that, since #P k ≥ 1 for all k, then H ≥ 0.

Definition 3.19. (Entropy)
If H = +∞, we define the entropy h, or the exponential rate of increasing information, by Interpretation of the amount of information H. The number #P k of different recurrent patterns Π k with length k measures the dynamical richness of N , with respect to the many possible finite words that the spiking-codes can show. Namely, H takes into account how many different clusters I n are exhibited at the milestones t n of the network. Thus, the total amount of information H is a quantitative index of the maximum global codified dynamical diversity that the network N is able to show along its recurrent evolution in the future, if the observer looks at each cell with an individual role. In fact, note that if i = j, then the event for which the cell i spikes and j does not, is distinguished from the converse event.

Theorem 3.20. (Amount of information)
Under the hypothesis of Theorem 3.5, the total amount of information H of the network is H = log 2 p ≤ log 2 1 + max{θ j : j ∈ N } min{∆ i,j : i, j ∈ N , i = j} .
We prove Theorem 3.20 in Section 5.

Definition 3.21. (Mutually similar cells)
We say that the cells of a fully cooperative network are mutually similar (with respect to the minimum interaction) if: and if the minimum positive interaction is large enough, i.e.: Now let us pose a result about large cooperative networks whose graphs are not necessarily complete.
and such that all the cells that do not belong to N 1 have non negative actions on the cells of N , i.e.
Corollary 3.24. (Networks with a large cooperative core.) Let N be a network that has a full cooperative core N 1 . Assume that the number m of cells in N 1 is large enough. Precisely: Then Assertions of Theorems 3.5, 3.6, 3.15, 3.20 and 3.22 hold, where min{∆ i,j : i, j ∈ N , i = j} must be substituted by min{∆ i,j : i ∈ N 1 , j ∈ N , i = j}.
We prove Corollary 3.24 in Section 8.
4. The proof of synchronization. In this section we prove Theorems 3.5 and 3.6. To do so, we need the following previous result: If from some initial state of the network there are two different (minimal) instants 0 < t * 0 < t * p when all the cells spike simultaneously, then the network eventually periodically synchronizes spikes with some natural spiking period p ≥ 1. (B) If from all the initial states the network eventually periodically synchronizes spikes with the same period p ≥ 1, then the number of all the possible recurrent code-patterns with length k ≥ 1 is and max k≥1 (#P k ) = p.
(C) If from some initial state the network eventually periodically synchronizes spikes with period p = 1, then, the sequence {I n } n≥n0 of clusters is constantly equal to the set of all the cells of the network, i.e.

(D)
If from all the initial states the network eventually periodically synchronizes spikes with period p = 1, then the number of all the possible recurrent code-patterns with length k ≥ 1 is Proof. Assertions (C) and (D) are immediate consequences of (A) and (B) respectively, in the particular case p = 1. Let us prove Assertion (A). Recall that m ≥ 2 denotes the number of cells in the network.
By hypothesis, all the cells of the network spike at instants t * 0 > 0 and t * p > t * 0 . Thus, due to the reset hypothesis in Definition 2.1, Formula (8), the state of the network at instant t * 0 , is x(t * 0 ) = x reset := x 1,reset , . . . , x i,reset , . . . , x m,reset , where x i,reset is the unique point in the state-space X i that satisfies Equality (6). (3) is deterministic, once fixed the unique state x reset of the network, a unique orbit exists in the future. Thus, translating the origin 0 of time to t * 0 , and recalling Equalities (3), we obtain:

As the dynamics defined by
) satisfies the differential equation (4) for all t > t * 0 such that t = t n , and S(Φ(x 0 , t * 0 )) = 0, we have: is the unique solution x * (·) = Φ(x reset , ·) of the deterministic dynamical system (3) with initial state x * (0) = x reset , plus the deterministic interactions' rule (10)during the time-interval (t * 0 , t * p ). By the hypothesis, all the cells spike simultaneously again at the instant t * p > t * 0 . Therefore, S(x(t * p )) = S(x(t * 0 )) = 0. Thus x(t * p ) = x(t * 0 ) = x reset . Then, . We deduce that the unique solution x * which has initial condition x reset , is periodic with period t * p − t * 0 . Thus, the instants t n and the spiking-codes I n are determined recursively from the unique periodic orbit x * . We deduce Also, and the sequence {t n } n≥0 of instants for which at least one cell spikes satisfies: Equalities (26), (27) and (28) with 0 ≤ r ≤ p − 1. In fact, for n = n 0 + hp + r ≥ p the code-pattern (I n , I n+1 , . . . , I n+k−1 ) coincides with (I n0+r , I n0+r+1 , . . . , I n0+r+k−1 ), because I n = I n0+hp+r = I n0+r . So, all the code-patterns in the list (29) are recurrent. Two or more codepatterns in the list (29) may coincide, so the number of different code-patterns with length k is at most equal to the number of items in the list (29). Thus: Now let us prove that, in the particular case that k = p, the code-patterns of the list (29) are pairwise different. To prove this assertion, with no loss of generality, we assume n 0 = 0 (if not, we translate the origin 0 of time to the instant t n0 ). Assume that (I r , I r+1 , . . . , I r+p−1 ) = (I s , I s+1 , . . . , I s+p−1 ) for 0 ≤ r, s ≤ p − 1. We must prove that r = s. The code-pattern I 0 = {1, . . . , m} will appear once in both patterns (30), because they both have length p, which is the (minimum) period of the sequence {I n } n≥1 . Say I 0 = I r+h = I s+k with 0 ≤ h, k ≤ p − 1.
Both patterns in Equality (30) coincide. Then the positions h and k are the same: h = k.
As I 0 = I r+h = I s+k , there are two indexes r + h and s + k, whose difference is smaller than p, such that the respective patterns coincide with I 0 = {1, . . . , m}. In other words, all the cells spike at two instants t r+h and t s+k such that |(r + h) − (s + k)| < p. But p is the minimum positive natural number such that all cells spike at instants t n and t n+p for some n. We deduce that r + h = s + k. Since we already know that h = k, we deduce that r = s, as wanted.
We conclude that all the code-patterns of the list (29) are pairwise different when the length k equals the period p. Thus, the number of different code-patterns with length p is p, ending the proof of Assertion (B) of Lemma 4.1.

Proof. Part a)
From Lemma 4.1, to prove that all the cells of the network eventually periodically synchronize spikes, it is enough to prove that there exist two instants 0 < t 0 < t p such that all the cells simultaneously spike at t 0 and at t p .
If for any initial condition we find a single instant t 0 > 0 at which all the cells simultaneously spike, then, taking as new initial state x(t 0 )) = x reset , we deduce that there exists a second instant t p > t 0 at which all the cells simultaneously spike. Thus, to prove Part (a) of Theorem 3.5, it is enough to show the following: Assertion (i) to be proved: For any initial condition, there exists an instant t 0 > 0 at which all the cells simultaneously spike.
From any initial state x 0 = x 1 (0), . . . , x m (0) such that 0 ≤ S i (x i (0)) < θ i for all 1 ≤ i ≤ m, consider the state and the m-dimensional vector whose components are the satisfaction variables S i (x i (t)) at instant t > 0.
Since each variable S i is governed by the differential equation (4) with a strictly positive real function g i : X i → R + (which is continuous on the compact space X i ), plus the eventual sum of interactions ∆ j,i ≥ 0 according to Equalities (10), we deduce: Property (ii) While no interference from outside the network appears, for each cell i the real variable S i is strictly increasing on t, for all t ≥ 0 such that S i (x i (t)) ∈ 0, θ i . Besides, except at those instants where S i is discontinuous, its derivative respect to time t exists, is positive and bounded away from zero, and at the instants where S i is discontinuous, the discontinuity jumps are positive.
We are assuming that S i (x i (0)) < θ i . Thus, from Property (ii), we deduce: Property (iii) For each cell i, there exists a first finite time t i > 0 such that lim t→t − i S i (x i (t)) = θ i . Namely, for any i ∈ {1, . . . , m} there exists a first spiking instant t i > 0.
Consider the minimum natural number K ≥ 1 such that From Inequalities (14) and (31), we deduce: By hypothesis, θ j > 0 for all 1 ≤ j ≤ m. Denote For later use, we note the following property: Assume that at the instant t > 0, at leat K cells are spiking, where K ≥ 1 is the natural number defined by Inequality (31). Then, for any other cell j ∈ {1, . . . , m}, applying Equalities (10), we have: Thus, any cell j will also spike at instant t, because its satisfaction variable S j (x j (t)) arrives to the goal level θ j . Summarizing, we have proved: Property (iv) If at least K cells spike at an instant t > 0, then all the cells spike at the instant t. Now, to end the proof of Assertion (i) it is enough to prove the following: Assertion (v) to be proved: There exists an instant T > 0 such that at least K cells spike simultaneously at T .
Let us take t 1 > 0 equal to the first positive instant when at least one cell i 1 arrives to its goal level, i.e.
Let us discuss according to two cases: either at least K cells spike at instant t 1 with the cell i 1 , or at most K − 1 do.
First Case. At least K cells spike at instant t 1 > 0. Thus Assertion (v) holds.
Second Case. At most K − 1 cells (and at least one cell i 1 ) spike at instant t 1 . From Inequality (32), there exist at least m − (K − 1) ≥ K 2 − (K − 1) ≥ 1 cells that do not spike at instant t 1 . Denote by A 1 this set of non spiking cells at instant t 1 . We have: Using Inequality (7), for each cell j ∈ A 1 we know that S j (x j (t − 1 )) ≥ 0. Since at instant t 1 at least the cell i 1 spikes, it sends a positive interaction ∆ i1,j to any cell j ∈ A 1 . Applying Inequalities (10) and (33), we deduce: In brief: (37) Denote by t 2 > t 1 the first instant after t 1 for which at least one cell i 2 arrives to its goal level. Now, we discuss again two cases: either there exist at least K cells that spike at instant t 2 with the cell i 2 , or there exists at most K − 1 cells that so do. In the first case, Assertion (v) holds. In the second case, denote by A 2 ⊂ A 1 the set of cells that did not spike at any instant in [0, t 2 ]. We have: Since t 2 > t 1 , applying Property (ii) and Inequalities (10) and (37), we obtain: Since at least one cell i 2 spikes at instant t 2 , it adds a positive jump ∆ i2,j to S j (x j (t − 2 )) for all j ∈ A 2 . Thus, using Inequality (33), we deduce: By induction on h ∈ N + , if 2 ≤ h ≤ K, assume that t h is the h-th. consecutive instant t h > t h−1 such that at least one cell i h spikes, and no more than K − 1 cells have spiked simultaneously at each instant t 1 < t 2 . . . < t h−1 . Denote by A h the set of cells that have not spiked at those instants and also do not spike at instant t h .
We have Arguing by induction as in Inequality (38), we obtain:

ELEONORA CATSIGERAS
For h = K, joining Inequality (40) and Equality (35), we deduce: In other words, we have proved that, if at each instant t 1 < t 2 < . . . < t K−1 , not more than K − 1 cells spike simultaneously, then, at instant t K the value of S j arrives to the goal level θ j , for any cell j ∈ A K . This implies that all the cells of A K spike simultaneously at instant t K > 0. Besides, from Inequality (39) there exist at least K 2 − K(K − 1) = K cells in the set A K . So, we have proved that at least K cells spike simultaneously at some instant t 1 > 0 or t 2 > t 1 or . . . t K−1 or t K . This ends the proof of Assertion (v), as wanted, and thus the proof of Part (a) of Theorem 3.5 is complete.

Proof. Part b)
Since the hypothesis (14) is a strict inequality, it establishes an open condition in the space of parameters of the network, endowed with the topology of Definition 2.5. Therefore, the Inequality (14) joint with any dynamical property that is deduced from it, is a robust phenomenon. In particular, due to Part (a) of Theorem 3.5, Inequality (14) joint with the eventually periodic synchronization of the spikes, is a robust phenomenon. This proves Assertion (b) of Theorem 3.5.

Proof of Theorem 3.6.
Proof. From the proof of Part (a) of Theorem 3.5, the waiting time T > 0 until the spike-synchronization of all the cells occurs, equals the time that takes the latest cell, say i, to arrive to its goal level θ i from its initial state x i (0). The worst case occurs when the initial state x i (0) of this slowest cell i is such that S i (x i (0)) takes its lowest possible value 0 (cf. Inequality (7)). Thus, to consider the worst case, we assume S i (x i (0)) = 0. Therefore T is smaller or equal than the time constant t i of the differential equation (4), for the solution S i (x i (t)) with initial state S i (x i (0)) = 0, such that S i (x i (t − i )) = θ i . This is because during the time-interval [0, t i ), some other cells j = i might have spiked. So they might have injected non negative jumps ∆ j,i to the instantaneous value of the variable S i of the cell i.
Then, the worst case if when all those jumps are zero. Summarizing, we have T ≤ t i and the worst case occurs if S i is only governed by the differential equation (4) for all t ∈ [0, t i ). Due to the mean value theorem of the differential calculus, there exists a time τ i ∈ [0, t i ) such that: Using the differential equation (4), we have: Joining Equality (41) and Inequality (42), we obtain: proving the first assertion of Theorem 3.6. Now, let us prove the second assertion of Theorem 3.6, finding an upper bound for the natural spiking period p. We revisit the proof of Part (a) of Theorem 3.5. We have defined the constant K as the minimum positive natural number that satisfies Inequality (31). Thus: On the one hand, by Property (iv) in the proof of Part (a) of Theorem 3.5, if K cells spike simultaneously at instant t i then all the cells spike simultaneously at instant t i . On the other hand, at the end of the proof of Assertion (v), (second case), we found that from any initial state, after at most K spikes of some cells (i.e. at instant t K as latest), there exist K cells that spike simultaneously. We conclude that, once all the cells had simultaneously spiked at instant t 0 , the minimum next instant t p > t 0 for which all the cells spike again simultaneously, is such that p ≤ K. Therefore: ending the proof of Theorem 3.6.
5. The proof of the amount of information. In this Section, we prove Theorem 3.20. We compute the total amount of information of fully cooperative networks with a large number m of cells.
Proof of Theorem 3.20. In Part (B) of Lemma 4.1 we have proved that where p is the natural spiking period, i.e the period of the sequence {I n } n∈N of spiking-codes. Therefore, from Formula (23), the total amount of information H that the network can generate or process is (The first equality in the above chain holds because the real function log 2 (x) is increasing on x ∈ R + .) We have proved that which is the first assertion of Theorem 3.20. Now, let us prove the second assertion. We use the upper bound of the natural spiking period p that was proved in Theorem 3.20, Formula (16). We conclude that ending the proof of Theorem 3.20.
6. The proofs of results on the protection factor. In this section we prove Proposition 3.14 and Theorem 3.15.

Proof of Theorem 3.15.
Proof. By hypothesis the network is fully cooperative. Thus, ∆ j,i > 0 for all j = i. By Equalities (19) and (21), the protection factor P i of the network to each cell is positive.
Since P i > 0, we obtain 1 − P i < 1. Applying Formula (22) of Lemma 3.14, and recalling that 0 < R i ≤ 1 -cf. Inequality (18)-we obtain: Therefore the net risk of death R i of the cell i when interacting in the network is strictly smaller than the intrinsic risk R i that i would have if it were not connected to the network. This ends the proof of Theorem 3.15.

7.
Proof of the results on networks with similar cells. In this section we prove Theorem 3.22: Proof. Part a) of Theorem 3. 22 From Part (a) of Theorem 3.5, there exists a first instant t 0 > 0 such that all the cells spike simultaneously at t 0 . So, to proof Part (a) of Theorem 3.22 it is enough to show the following assertion, under the additional hypothesis of Inequality (24): Assertion (vi) to be proved. If at some instant t 0 > 0 all the cells spike simultaneously, and if t 1 > t 0 is the first instant after t 0 when at least one cell spikes, then all the cells spike simultaneously at t 1 .
Fix i, one of the cells that spike at instant t 1 > t 0 . By hypothesis, the cell i has also spiked at instant t 0 , but not during the inter-spike-interval (t 0 , t 1 ). Then, applying the reset rule (8), we have: Since i also spikes at instant t 1 > t 0 , we have: No cell spikes during the time-interval (t 0 , t 1 ). Thus, S i is governed by the differential equation (4) during such a time-interval. Applying the mean value theorem of the differential calculus, there exists τ i ∈ (t 0 , t 1 ) such that: Using the differential equation (4), and recalling that x i ∈ X i , X i is compact and g is continuous, we obtain: Joining Equality (43) and Inequality (44), we deduce: Now denote by j any cell. It spikes at instant t 0 . Then, No cell spikes during the time-interval (t 0 , t i ). Thus, S j is governed by the differential equation (4) during such an interval of time. Applying the mean value theorem of the differential calculus, there exists τ j ∈ (t 0 , t 1 ) such that: From the differential equation (4), we deduce: Joining Inequalities (46) and (47), we deduce: Joining Inequalities (45) and (48), we obtain: By hypothesis, Inequality (24) holds. So, the factor at right (between large parenthesis) in Inequality (49) is bounded from below by −(min i =j ∆ i,j )/ max 1≤i≤m θ i . We deduce: Since at instant t 1 the cell i spikes, it sends an action ∆ i,j to each cell j. From Inequality (50), and from the interaction rule in Equalities (10), we get: Therefore, the variable S j of any cell j arrives to its goal level θ j at the instant t 1 . Thus, any cell j spikes at t 1 , and Assertion (vi) is proved. This ends the proof of Part (a) of Theorem 3.22.
Proof. Part b) of Theorem 3.22 First, let us prove that the protection factor P i of the network to each cell i, according to Definition 3.12, satisfies min{1, P i } = 1 (to be proved). (51) In fact, by Formula (19) the net sum of the actions ∆ (h) i,N that the cell i receives from the other cells of the network during its h-th. inter-spike-interval (t where m is the number of cells of the network. By hypothesis, Inequality (14) holds. Therefore m ≥ 3, which implies m − 1 ≥ √ m. We deduce that: Substituting in (53), we obtain: ∆ Second, we apply Lemma 3.14. From Equalities (22) and (51), and Inequality (18), we deduce: Proof. Part c) of Theorem 3. 22 From Part (a) of Theorem 3.22, under the additional hypothesis stated by Inequality (24), the period p of the sequence {I n } n∈N of spiking-codes is p = 1. From Theorem 3.20, we know that the total amount of information H that the network can generate of process is log 2 p. Therefore, H = log 2 p = log 2 1 = 0, as wanted.

Proof of Corollary 3.24. (Large full cooperative core)
Proof. Since by hypothesis, the number m of cells of the core N 1 satisfies Inequality (25), we can repeat all the arguments in the proof of Part (a) of Theorem 3.5 in Subsection 4.1, by substituting the network N by the core N 1 . So, we deduce that there exists a strictly increasing sequence {t n } n≥0 of instants t n → +∞, and a natural period p ≥ 1, such that: At least one cell of N 1 spikes at each instant t n , for all n ≥ 0.
No cell of N 1 spikes in the open time-intervals (t n , t n+1 ). All the cells of N 1 spike simultaneously at the instant t hp for all h ≥ 0. Therefore, to prove the eventual periodic synchronization of the whole network N , it is enough to prove that the cells of N spike altogether at each instant t such that all the cells of the core N 1 spike.
In fact, we repeat, with slight changes, the proof of Property (iv) in Subsection 4.1: We define the minimum natural number K ≥ 1 such that K > max{θ j : j ∈ N } min{∆ i,j : i ∈ N 1 , j ∈ N , i = j} .
Due to Inequality (25), the number m of the core N 1 satisfies √ m > K.
Now, we repeat the same arguments of the proof in Subsection 4.1, starting at Equality (33) and ending just before Property (iv): Substituting min i =j ∆ i,j by min{∆ i,j : i ∈ N 1 , j ∈ N , i = j}, we deduce Property (iv)': If at least K cells of the core N 1 spike at instant t > 0, then all the cells of the network N spike at instant t.
Since at instant t hp all the m cells of the core N 1 spike, and m ≥ 3 satisfies Inequality (54), we obtain: m > √ m > K.
We conclude that the cells of the network spike altogether at instants t hp for all h ≥ 0. Thus, the whole network periodically synchronizes spikes from any initial state (after a finite transitory time). Once we have proved the periodic synchronization of the network, the proof of the other assertions under the hypothesis of Corollary 3.24, follow the same arguments in the proofs of Sections 4, 5 and 6, after substituting min i =j ∆ i,j by the expression (55).