Multiple Phase Transitions for an Infinite System of Spiking Neurons
aa r X i v : . [ m a t h . P R ] F e b Multiple Phase Transitions for an Infinite Systemof Spiking Neurons
A. M. B. Nascimento ∗ †
Abstract
We consider a stochastic model describing the spiking activity of a countableset of neurons, which are spatially organized into a homogeneous tree of degree d , d ≥
2. Roughly, the model is as follows. Each neuron is represented by its membranepotential, which assumes non-negative integer values. Neurons spike at Poisson rateone, provided they have strictly positive membrane potential. When a spike occurs,the spiking neuron’s potential changes to 0, and all neurons connected to it receivea positive amount of potential. Moreover, between successive spikes and withoutreceiving any spiking inputs from other neurons, each neuron’s potential behavesindependently as a pure death process with death rate γ ≥
0. In this article, we showthat if the degree d is large enough, then the process exhibits at least two phasetransitions depending on the choice of rate γ : For large values of γ , the neuralspiking activity goes extinct almost surely; For small values of γ , a fixed neuronspikes infinitely many times with positive probability, and for “intermediate” valuesof γ , the system has a positive probability of always presenting spiking activity,but this activity drifts off to infinity, and each neuron eventually stops spiking andremains at rest forever. AMS 2010 Mathematics Subject Classification.
Key words and phrases. system of spiking neurons, multiple phase transition, trees
We consider in this article a simple stochastic model, which is a continuous-time Markovprocess denoted by ξ t , describing the spiking activity within an infinite network of neurons.Informally, the model’s definition is as follows. Let us consider a graph G = ( V, E ) torepresent the network of neurons; vertices in V represent neurons and edges in E indicatethe existence of a connection between them. For each t ≥
0, the random variable ξ t ,which takes values in Z V + , where Z + = { , , . . . } , is a configuration giving the membranepotential for all neurons in V , that is, ξ t ( i ) ∈ Z + stands for the membrane potential of ∗ Partially supported by CNPq grant 155972/2018-9 † Centro de Ciências Exatas e da Terra, Universidade Federal do Rio Grande do Norte, Av SenadorSalgado Filho 3000, Campus Universitário de Natal, 59078-970 Natal RN, Brasil. (email: [email protected]) i at time t . This configuration evolves in time with transitions governed by thefollowing rules. To each neuron we assign a Poisson process of rate one in such a way thatwhen the exponential time occurs at an active neuron (that is to say, a neuron with strictlypositive membrane potential), that neuron spikes. As a result, the potential of the spikingneuron is immediately changed to a resting value, which is assumed to be 0, and at thesame time all neurons in the network which are connected to it increase their potentialsby a positive value; this value may vary for each pair of neurons. Furthermore, betweentwo consecutive spikes and without receiving any spiking inputs from other neurons, thepotential of each neuron behaves independently as a pure death process with death rate γ ≥ leakageeffect , a natural phenomenon that causes neurons to spontaneously lose potential. Therate γ will be, therefore, called the leakage parameter . In this article, we shall assume thatneurons are spatially organized into infinite trees, which are roughly speaking connectedgraphs with undirected edges and no cycles. A rigorous description of this model will begiven in the next section.Since the neuron membrane potential vanishes each time it spikes, this process can beseen as an extension to the continuous-time framework of the model introduced by Galvesand Löcherbach in [10] (see also [11] for a critical review), where neurons are representedthrough their spike trains. There are many other versions, in discrete and continuoustime, of the latter model in the literature; let us mention [3, 5–9, 18]. In fact, our modelis quite close to a particular version of the process considered by Ferrari et al. [8], in thatneurons spike at Poisson rate one whenever they have a strictly positive potential. Wepoint out, however, that in our model the potential gain induced by spikes is not supposedto be uniform. In addition, we consider the leakage effect differently, as the membranepotential decreases by only one unit each time it occurs. We remark, on the other hand,that that particular model in [8] can be seen as a variant of ours with the same leakageparameter, but for which we do not allow neurons to accumulate more than one unit ofpotential. We will also be interested in this variant of the model in the present article .Our motivation for this project comes in part from that work of Ferrari et al. in [8].In that paper, they proved the existence of a phase transition situation for the neuralspiking activity, which was interpreted as a sudden change of behavior in the number ofspikes emitted by every neuron as a function of the leakage parameter. Specifically, bytaking the network of neurons as the one-dimensional integer lattice Z (equipped withits usual graph structure), they found a distinguished value γ , 0 < γ < ∞ , for theleakage parameter for which the following holds: If the process starts from a potentialconfiguration in which all neurons are active, then there exists a positive probabilitythat every neuron spikes infinitely many times for γ < γ , while that for γ > γ , everyindividual neuron eventually stops spiking and remains at rest forever. These behaviorsare usually called in the statistical physics literature local survival and local extinction ,respectively, and so γ can be seen as the critical point corresponding to the local survival(or extinction) of the spiking activity in Z . From this picture, a natural question emergeswhen the concept of global survival comes into play. By global survival we mean theexistence of a critical point 0 < γ < ∞ , so that the neural spiking activity goes extinctwith probability 1 if γ is above γ and the system has a positive probability of alwayspresenting some spiking activity if γ is below γ . Clearly, local survival implies globalsurvival; therefore γ ≤ γ . We wonder if γ < γ . When this strict inequality holds theprocess has two phase transitions.The purpose of this article is thus to prove that we have these two phase transitions forthe presented process when it starts from a state having a single active neuron. We shall2nly consider homogeneous trees to represent the network of neurons. A homogeneoustree is the unique connected tree in which every vertice has degree d , d ≥
2, where thedegree of a vertice is the number of connections it has. On this structure, we show thatthe process exhibits two phase transitions provided the number d is large enough. Inaddition, we present some estimates for the expectation of the total membrane potentialwithin the system at any instant of time.Before finishing this introduction, let us briefly point out some of our motivations fortaking neurons spatially organized into trees. Firstly, some theoretical works on systemsof spiking neurons have considered the underlying graph as a slightly supercritical Erdös-Rényi random graph (see e.g. [10]), commonly based on arguments by Beggs and Plenz [1],who argue that networks of living neurons should behave in a slightly supercritical state.However, it has been shown in [14] that the neighborhood of random points in a slightlysupercritical Erdös-Rényi graph looks like the neighborhood of a fixed vertice in a tree.Therefore, trees seems to be as likely as Erdös-Rényi random graphs to represent, at leastlocally, real networks of neurons. Another reason is that the main purpose of the currentarticle concerns the existence of multiple phase transitions for the presented process and,as suggested by Pemantle’s work [17] on the contact process, trees are probably the moststraightforward graph structure on which we can observe such behavior.The remainder of the article is organized as follows. In Section 2 below, we presentthe general model, introduce some notation, and state our main result, Theorem 1, whoseproof follows easily from Theorem 2 on bounds for the critical values corresponding tolocal and global survival. In Section 3, we prove Theorem 2 and discuss some implicationsof its proof for a particular version of the model introduced by Ferrari et al. in [8]. We now give a general definition of the process we study. To do so we need the followingingredients:(i) A graph G = ( V, E ), consisting of a set of vertices V and a set of (undirected)edges E , E ⊂ V , to represent the network of neurons. The graph G is assumed to beconnected and of bounded degree, where the degree of a vertice (neuron, from now on) isjust the number of connections it has.(ii) A matrix w = ( w i → j ) i,j ∈ V with non-negative integer entries to measure the po-tential gain induced by spikes; w i → j will be the value added to the potential of neuron j when neuron i spikes. We assume here that for each i ∈ V , w i → j = 0 unless j ∈ N i ,where N i = { j ∈ V : { i, j } ∈ E } is the interaction neighborhood of i in G . Moreover, weimpose that λ := sup i ∈ V X j ∈N i w i → j < ∞ . (2.1)(iii) A parameter γ ≥ γ the leakage param-eter .For a fixed w , the model we consider is thus defined as the continuous-time Markov process ξ t = ( ξ t ( i ) , i ∈ V ) on G with state space being S = n ξ ∈ Z V + : P i ∈ V ξ ( i ) < ∞ o , ξ is a configuration giving the state of the membrane potential for the set V of allneurons), whose generator is given, for any cylindrical function, by Lf ( ξ ) = X i ∈ V { ξ ( i ) ≥ } ( f (∆ ∗ i ξ ) − f ( ξ )) + γ X i ∈ V ξ ( i ) (cid:16) f (∆ † i ξ ) − f ( ξ ) (cid:17) , (2.2)where for each ξ ∈ S such that ξ ( i ) ≥
1, ∆ ∗ i ξ is given by(∆ ∗ i ξ )( j ) = ξ ( j ) + w i → j , if j ∈ N i , = ξ ( j ) , if j ∈ V \ ( N i ∪ { i } ) , = 0 , if j = i, (2.3)and for each ξ ∈ S , ∆ † i ξ is defined as(∆ † i ξ )( j ) = ξ ( j ) − , if j = i, = ξ ( j ) , if j = i. (2.4)An interpretation of (2.2) is as follows. Its first term describes the neural spikingactivity within the network: If neuron i is active, when it has strictly positive membranepotential, then at rate one it spikes, instant at which its potential is reset to 0 andsimultaneously every neuron j belonging to its interaction neighborhood N i adds theamount of w i → j to its potential. The second term models the spontaneous loss of potentialthat neurons are subject to within the system due to the leakage effect: If neuron i hasmembrane potential equals k , k ≥
1, then after an exponential time of rate γk , itspotential decreases to k − ξ t using a graph techniquecalled graphical representation [13]. For that sake, we associate to each neuron i ∈ V independent Poisson processes N ∗ i of rate one and N † i,k of rate γ for k ≥
1. The N ∗ i process will give the times at which neuron i spikes, and the N † i,k processes will give thetimes at which neuron i is affected by the leakage. The process construction is done asfollows. Start with finitely many neurons with strictly positive potential. At an arrivaltime of N ∗ i , if ξ t ( i ) ≥
1, we replace its value by 0 and the value of ξ t ( j ) by ξ t ( j ) + w i → j forall j ∈ N i . At an arrival time of N † i,k , we replace the value of ξ t ( i ) by ξ t ( i ) − k ≤ ξ t ( i ).Along this article we focus on the case G = T d , where T d is the homogeneous treewith degree d , d ≥
2. (Note that T = Z , the one-dimensional integer lattice). In doingso, it will be convenient to think of T d arranged into levels in such a way that somedistinguished neuron, denoted by o and called the root , is in level 0 and every neuron inlevel n ∈ Z has exactly one neighbor (its parent) in level n − n + 1. We shall use l ( i ) to indicate the level of neuron i ∈ V . Forfuture reference, let us set w = sup i ∈ V { w i → j : l ( j ) = l ( i ) − } and w = sup i ∈ V { w i → j : l ( j ) = l ( i ) + 1 } . (2.5)Evidently all these quantities are finite.We shall denote by ξ ξt the process starting from a potential configuration ξ ∈ S attime 0. When ξ equals e o , where e o ( i ) = 1 , if i = o ;= 0 , if i = o, (2.6)4e shall write ξ e o t (or ξ e o ,γt when needed to make explicit the leakage parameter in thenotation).Let | ξ e o t | = P i ∈ V ξ e o t ( i ) be the total membrane potential present within the network attime t . We shall say that ξ e o t survives (globally) if P w ( | ξ e o t | ≥ , ∀ t ≥ > P w stands for the law of the process with fixed matrix w . When ξ e o t does notsurvive, we say it dies out . If P w ( ξ e o t ( o ) ≥ > ξ e o t survives locally . When the process survives globally but does notsurvive locally we shall say it survives weakly . We remark that since T d is a connectedgraph the definition of local survival does not depend on the particular neuron o . Wealso remark that, in view of Lemma A.1 in Appendix A, the probability of global survival P w ( | ξ e o ,γt | ≥ , ∀ t ≥
0) is a non-increasing function of parameter γ .Now given w let γ l and γ g be the following critical values for the leakage parameter γ : γ l = sup { γ : P w ( ξ e o ,γt ( o ) ≥ > } ,γ g = sup { γ : P w ( | ξ e o ,γt | ≥ , ∀ t ≥ > } . (2.7)In words, γ l is the critical value corresponding to the local survival and γ g is the criticalvalue corresponding to the global survival of the spiking activity in T d . Clearly, γ l ≤ γ g .We now state our main result, which presents sufficient conditions to have γ l < γ g ,and hence two phase transitions for the process. Theorem 1.
We have γ l < γ g (two phase transitions for ξ e o t ) whenever d > w w + 1 . As a particular conclusion of this result, we have the following.
Corollary 2.1.
If the matrix w is such that for each i ∈ V , w i → j = { j ∈ N i } , then γ l < γ g for all d ≥ . Theorem 1 follows immediately from Theorem 2 stated below. In fact, our strategy toproving it is standard and consists in deducing an upper bound for γ l and a lower bondfor γ g which are good enough to imply γ l < γ g if d is sufficiently large. Theorem 2.
We have the following bounds for the two critical values: γ l ≤ q w w ( d − − and d − ≤ γ g ≤ λ − Remark 2.1.
It is important to note that, as λ ≥ , the bounds in (2.9) readily implythat, for all d ≥ , the process ξ e o t has at least one phase transition. The proof of Theorem 2 is done in Section 3 next. The strategy is as follows. Theupper bounds in (2.8) and (2.9) are both obtained based on a method employed byLiggett [16] (see Theorem 4.8. therein) to compute the exact values of the two criticalpoints for the branching random walk on homogeneous trees. Generally speaking, themethod consists basically in relating global and local survival of the process to propertiesof an auxiliary process specified by a convenient weighting function defined on S . To getthe lower bound in (2.9), we apply a coupling argument that compares our process to anon-spatial branching process, which is constructed by using the graphical representationof the process and exploring the nice structure (no cycles) of trees.5 Bounds on the Critical Values – Proof of Theo-rem 2
Let ν ρ : S → [0 , ∞ ) be a weighting function defined by ν ρ ( ξ ) = X i ∈ V ξ ( i ) ρ l ( i ) , (3.1)where ρ is a non-negative parameter to be specified later. Note that ν ( ξ ) = | ξ | .In what follows, we shall consider and study the auxiliary process ν ρ ( ξ ηt ) in order todeduce upper bounds for γ l and γ g . Our key ingredient will be the following lemma. Lemma 3.1.
For any configuration η ∈ S and any t ≥ , we have E w ( ν ρ ( ξ ηt )) ≤ ( φ γ ( ρ )) t ν ρ ( η ) , (3.2) where E w stands for the expectation with respect to the probability measure P w , and φ γ ( ρ ) = exp ( λ − − γ ) , if ρ = 1;= exp (cid:16) w ρ − + w ( d − ρ − − γ (cid:17) , if ρ = 1 . (3.3) Remark 3.1.
Observe that for any fixed γ , φ γ ( ρ ) admits a unique global minimum, whichis attained at point ρ ∗ = q w / w ( d − .Proof of Lemma 3.1. Recall the definitions (2.3) and (2.4). From the definition (2.2), wehave that Lν ρ ( ξ ) = − γ X i ∈ V ξ ( i ) ρ l ( i ) + X i ∈ V { ξ ( i ) ≥ } X j ∈N i w i → j ρ l ( j ) − ξ ( i ) ρ l ( i ) ≤ − (1 + γ ) ν ρ ( ξ ) + X i ∈ V ξ ( i ) X j ∈N i w i → j ρ l ( j ) (3.4)where the inequality follows from the fact that { ξ ( i ) ≥ } ≤ ξ ( i ), for all i ∈ V . Now set c ρ = λ, if ρ = 1;= w ρ − + w ( d − ρ, if ρ = 1 , and note that P j ∈N i w i → j ρ l ( j ) ≤ c ρ ρ l ( i ) , for all i ∈ V . Thus, Lν ρ ( ξ ) ≤ ( c ρ − − γ ) ν ρ ( ξ ) , and by classical results on Markovian generators, ddt E w ( ν ρ ( ξ ξt )) ≤ ( c ρ − − γ ) E ( ν ρ ( ξ ξt )) , so that (3.2) is now just a matter of applying Grönwall’s lemma in the above inequality.From Lemma 3.1 we readily have that Proposition 3.1. E w ( | ξ e o t | ) ≤ exp ( λ − − γ ) t . (3.5) In particular, ξ e o t dies out for any γ ≥ λ − . Therefore, γ g ≤ λ − . roof. First, note that (3.5) is just inequality (3.2) set up with ξ = e o and ρ = 1. Now,let us assume that γ > λ −
1. Since in this case E w ( | ξ e o t | ) → t → ∞ , the classicalMarkov inequality readily implies P w ( | ξ e o t | ≥ , ∀ t ≥
0) = lim t →∞ P w ( | ξ e o t | ≥
1) = 0 . (3.6)Therefore, ξ e o t dies out for any γ > λ − γ = λ −
1. Indeed, since (3.5)implies lim t →∞ E w ( | ξ e o t | ) ≤ , (3.7)if ξ e o t survives at γ = λ −
1, then Lemma A.2 in Appendix A would imply that (almostsurely) | ξ e o t | >
1, for all large t . This contradiction finishes the proposition’s proof. Remark 3.2.
Observe that Proposition 3.1 still holds if, instead of a homogeneous tree,the network of neurons is a connected graph of bounded degree. This is so because theargument used to establish inequality (3.2) when ρ = 1 requires only that λ < ∞ and,by definition, this condition holds if the graph representing the network of neurons hasbounded degree. Our next step in proving Theorem 2 concerns the critical point γ l . We are going toprove that Proposition 3.2. γ l ≤ q w w ( d − − .Proof. Suppose γ > q w w ( d − −
1, and set M t = ν ρ ∗ ( ξ e o t )( φ γ ( ρ ∗ )) t , where ρ ∗ = w / w ( d − . Let F t = σ ( ξ e o s : s ≤ t ). By the Markov property and inequal-ity (3.2) we may find that E w ν ρ ∗ ( ξ e o t + s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t ! = E w (cid:16) ν ρ ∗ ( ξ ξ eot s ) (cid:17) ≤ ( φ γ ( ρ ∗ )) s ν ρ ∗ ( ξ e o t ) , ∀ s, t ≥ . (3.8)Now, using that φ γ ( ρ ∗ ) < γ > q w w ( d − −
1, we readily get from (3.8)that M t is a supermartingale and, since it is non-negative, it converges almost surely (seee.g. [4], p. 351). This together with the fact that ( φ γ ( ρ ∗ )) t → t → ∞ , implies that ν ρ ∗ ( ξ e o t ) → t → ∞ , and so ξ e o t can not survive locally. Therefore, γ l ≤ q w w ( d − − γ g ≥ d −
2. To do so we shall consider an auxiliary Markovprocess η t = ( η t ( i ) , i ∈ V ) on network T d and taking values in { , } V in which neurons atstate 1 are regarded as active , meaning they have membrane potential larger than 0, whileneurons at state 0 are interpreted as quiescent , indicating they have membrane potential0. This auxiliary process is defined as follows. Active neurons spike (that is, they becomequiescent and make at the same time all of their neighboring neurons active) at rate 1, orthey become spontaneously quiescent at rate γ . Quiescent neurons become active only if7t least one of their active neighboring neurons, if there are any, spikes. In what follows,we shall denote by η ηt the process η t starting from η = η ∈ { , } V .We notice that this process can be seen as a version of our model with the same leakageparameter, but for which we do not allow neurons to accumulate more than one unit ofpotential. This nice point of view is key in our argument since it leads to a monotonecoupling that will be needed in the next proposition: For fixed w , using the graphicalrepresentation we can construct η t and ξ t simultaneously, but for the process η t we useonly the Poisson processes N ∗ i and N † i, , in such a way that η ( i ) ≤ ξ ( i ) , ∀ i ∈ V implies η ηt ( i ) ≤ ξ ξt ( i ) , ∀ i ∈ V. (3.9)With this property, we now can prove that Proposition 3.3. γ g ≥ d − .Proof. For fixed w , (3.9) readily implies that P ( | η e o t | ≥ ≤ P w ( | ξ e o t | ≥
1) (3.10)for any t ≥
0. Hence, P ( | η e o t | ≥ , ∀ t ≥
0) = lim t →∞ P ( | η e o t | ≥ ≤ lim t →∞ P w ( | ξ e o t | ≥
1) = P w ( | ξ e o t | ≥ , ∀ t ≥ . So it is enough to show that the process η e o t survives whenever γ < d −
2. We do thisas follows. Let ζ t be a process on T d which evolves according the following rules. Startwith only one active neuron, the root. It waits a mean 1 exponential time at the end ofwhich it spikes, becoming quiescent and making simultaneously all of its neighbors exceptits parent active. In general, each new active neuron make active, when it spikes, all ofits neighboring neurons except the parent one. This restriction is to ensure that once aneuron is active, and then becomes quiescent, it will never become active again. Neuronsbecome quiescent if they spike or when the leakage occurs, the latter happening at rate γ , independently.From the above rules, it is not difficult to see that we can construct ζ t and η e o t simul-taneously in such a way that | ζ t | ≤ | η e o t | for all t ≥
0. Furthermore, since T d have nocycles, | ζ t | , which is just the number of active neurons in ζ t at time t , defines a continuous-time (non-spatial) branching process with offspring distribution given by p (0) = γ γ and p ( d −
1) = γ . Now, as it is well-known, branching processes survive if and only if themean offspring distribution is strictly greater than 1. Thus, if d − γ > γ < d −
2, we have that the process | ζ t | survives, and so does | η e o t | . Therefore, γ g ≥ d − Remark 3.3.
Notice that, in view of Proposition 3.1, using the coupling argument estab-lishing Proposition 3.3 above, we may prove the following statement: For any d ≥ , exp ( d − − γ ) t ≤ E w ( | ξ e o t | ) ≤ exp ( λ − − γ ) t . (3.11) Application.
We conclude this section with an application of some of the above resultsto a particular case (on T d ) of the system of spiking neurons introduced by Ferrari etal. [8], which is essentially equivalent to the auxiliary Markov process η t introduced in the8aragraph following the proof of Proposition 3.2 (see Section 3 in [8]). Let γ and γ bethe two critical values of η e o t : γ ≡ γ ( T d ) = sup { γ : P ( η e o t ( o ) ≥ > } ,γ ≡ γ ( T d ) = sup { γ : P ( | η e o t | ≥ , ∀ t ≥ > } . Now consider the process ξ e o t having matrix w = ( w i → j ) i,j ∈ V with entries given by w i → j = { j ∈ N i } , for each i ∈ V . From (3.9) and Propositions 3.1 and 3.2, we get that γ ≤ γ l ≤ √ d − − γ ≤ γ g ≤ d −
1. This together with the bound γ ≥ d − γ < γ (the process η e o t has two phase transitions) for all d ≥
6, and so does the particular system(on T d ) considered by [8]. A Appendix
Lemma A.1. If γ ′ < γ ′′ , then for any t ≥ we have P w (cid:16) | ξ e o ,γ ′ t | ≥ (cid:17) ≥ P w (cid:16) | ξ e o ,γ ′′ t | ≥ (cid:17) . (A.1) Proof.
It is enough to construct the processes ξ e o ,γ ′ t and ξ e o ,γ ′′ t simultaneously in such away that ξ e o ,γ ′′ t ( i ) ≤ ξ e o ,γ ′ t ( i ) , ∀ i ∈ V, ∀ t ≥ . (A.2)To achieve this we proceed as follows. First we define both processes using the samePoisson processes N ∗ i of rate one. Then, we consider independent Poisson processes N † ,γ ′′ i,k of rate γ ′′ for k ≥ γ ′ / γ ′′ to obtain the right leakage rate for ξ e o ,γ ′ t : withprobability γ ′ / γ ′′ an event time of N † ,γ ′′ i,k is accepted and the leakage may also occur in ξ e o ,γ ′ t . Lemma A.2. On Ω ∞ := {| ξ e o t | ≥ , ∀ t ≥ } , we have that | ξ e o t | → ∞ almost surely.Proof. The result is proven arguing as Lemma 2.1. in [2] using however that, if | ξ e o t | ≤ N for some integer N ≥
1, then the probability that all membrane potential within thenetwork will vanish before any spiking is at least ( γ / γ ) N . Acknowledgements
This article is part of the postdoctoral project (CNPq grant 155972/2018-9) of the authorat IME-USP and has been produced as part of the activities of FAPESP Research, In-novation and Dissemination Center for Neuromathematics (grant 2013/07699-0, S. PauloResearch Foundation). The author would like to thanks Antonio Galves, Aline Duarteand Daniel Takahashy for stimulating discussions about this subject.
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