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We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework.
The aim of this paper is to prove Poincaré inequalities for the semigroup , as well as for the invariant measure, of the model introduced in  by Galves and Löcherbach, to describe the activity of a biological neural network. What is particularly interesting about the jump process in question is that it is characterized by degenerate jumps, in the sense that after a particle (neuron) spikes, it loses its memory by jumping to zero. Furthermore, the probability of a spike of a particular neuron at any time depends on its actual position and thus the past of the whole neural system.
For the associated semigroup, first we prove some Poincaré-type inequalities of the form
for any possible starting configuration x. We give here the general form of the type of inequalities investigated in this paper; however to avoid overloading this introduction with technical details we postpone the definitions of classical quantities such as the “carré du champ” and other notations used here to Section 1.3.
Then, we restrict ourselves to the special case where the initial configuration x is in the domain of the invariant measure, and we derive the stronger Poincaré inequality
Then we show a Poincaré inequality for the invariant measure
Before we describe the model, we present the neuroscience framework of the problem.
1.1. The Neuroscience Framework
The activity of one neuron is described by the evolution of its membrane potential. This evolution presents from time to time a brief and high-amplitude depolarization called an action potential or spike. The spiking probability or rate of a given neuron depends on the value of its membrane potential. These spikes are the only perturbations of the membrane potential that can be transmitted from one neuron to another through chemical synapses. When a neuron i spikes, its membrane potential is reset to 0 while the so-called “post-synaptic neurons” influenced by neuron i receive an additional amount of membrane potential.
From a probabilistic point of view, this activity can be described by a simple point process since the whole activity is characterized by the jump times. In the literature, Hawkes processes are often used to describe systems of interacting neurons, see [1,2,3,4,5,6] for example. The reset to 0 of the spiking neuron provides a variable length memory for the dynamic and therefore point processes describing these systems are non-Markovian.
On the other hand, it is possible to describe the activity of the network with a process modeling not only the jump times but the whole evolution of the membrane potential of each neuron. This evolution needs then to be specified between the jumps. In  the process describing this evolution follows a deterministic drift between the jumps, more precisely the membrane potential of each neuron is attracted with exponential speed towards an equilibrium potential. This process is then Markovian and belongs to the family of Piecewise Deterministic Markov Processes introduced by Davis ([8,9]). Such processes are widely used in probability modeling of e.g., biological or chemical phenomena (see e.g.,  or , see  for an overview). The point of view we adopt here is close to this framework, but we work without drift between the jumps. We therefore consider a pure jump Markov process and will make use of the abbreviation PJMP in the rest of the present work.
We consider a process where N is the number of neurons in the network and where for each neuron and each time each variable represents the membrane potential of neuron i at time . Each membrane potential takes value in . A neuron with membrane potential x “spikes” with intensity where is a given intensity function. When a neuron i fires, its membrane potential is reset to interpreted as resting potential, while the membrane potential of any post-synaptic neuron j is increased by . Between two jumps of the system, the membrane potential of each neuron is constant.
Working with Hawkes processes allows consideration of systems with infinitely many neurons, as in  or . For our purpose, we need to work in a Markovian framework and therefore our process represents the membrane potentials of the neurons, considering a finite number N of neurons.
1.2. The Model
Let be fixed and be a family of i.i.d. Poisson random measures on with intensity measure . We study the Markov process taking values in and solving, for , for ,
In the above equation for each is the synaptic weight describing the influence of neuron j on neuron . Finally, the function is the intensity function.
This can be seen in the following way. The first term is the starting point of the process at time . The second term corresponds to the reset to 0 of neuron i when it spikes. The point process gives the times where a spike of neuron i can occur which actually happens with rate leading to a reset to 0 of neuron i due to the term . The third term corresponds to the modification of membrane potential of post-synaptic neurons influenced by neuron The modification value is given by the synaptic weight provided the new value is smaller than the maximum potential which is ensured by the second indicatrix function.
The generator of the process X is given for any test function and is described by
for some . With this definition the process remains inside the compact set
Furthermore, we also assume the following conditions about the intensity function:
for some strictly positive constants c and .
The probability for a neuron to spike grows with its membrane potential so it is natural to think of the function as an increasing function. Condition (5) implies that this growth is at least linear, and models the spontaneous activity of the system: whatever the configuration x is, the system will always have a positive spiking rate.
1.3. Poincaré-Type Inequalities
Our purpose is to show Poincaré-type inequalities for our PJMP, whose dynamic is similar to the model introduced in . The main difference between our framework and the one of  relies in the fact that as in , we study a process modeling the membrane potential of each neuron instead of a point process focusing only on the spiking events. Focusing on the spike train is sufficient to describe the network activity since it contains all the relevant information. However, the membrane potential integrates each relevant spike that occurred in the past and this gives us a Markovian framework. Regardless of the point of view, the notable difference in dynamics with  is the absence of drift between jumps. We assume here that there is no loss of memory between spikes. Therefore, our conclusions cannot directly apply to the model studied in .
We will investigate Poincaré-type inequalities at first for the semigroup and then for the invariant measure . Concerning the semigroup inequality, we will study two different cases. The first, the general one, where the system starts from any possible initial configuration. Then, we restrict to initial configurations that belong to the domain of the invariant measure.
Let us first describe the general framework and define the Poincaré inequalities on a discrete setting (see also [13,14,15,16,17]). At first we should note a convention we will widely use. For a function f and measure we will write for the expectation of the function f with respect to the measure that is
We consider a Markov process which is described by the infinitesimal generator and the associated Markov semigroup . For a semigroup and its associated infinitesimal generator we will need the following well know relationships: (see for example ).
We define to be the invariant measure for the semigroup if and only if
Furthermore, we define the “carré du champ” operator (see ) by:
For more details on this important operator and the inequalities that relate to it one can look at [18,19,20]. For the PJMP process defined above with the specific generator given by (2) a simple calculation shows that the carré du champ takes the following form.
We say that a measure satisfies a Poincaré inequality if there exists constant independent of f, such that
where the variance of a function f with respect to a measure is defined with the usual way as: . It should be noted that in the case where the measure is the semigroup , then the constant C may depend on t, .
For the Poincaré inequality for continuous time Markov chains one can look in [16,21]. In [13,14,17], the Poincaré inequality (SG) for has been shown for some point processes, for a constant that depends on time t, while the stronger log-Sobolev inequality, has been disproved. The general method used in these papers that will be followed also in the current work, is based on the so-called semigroup method which shows the inequality for the semigroup .
The main difficulty here is that for the pure jump Markov process that we examine in the current paper, the translation property
used in [13,17] does not hold here. This appears to be important because the translation property is a key element in these papers, since it allows to bound the carré du champ by comparing the mean where the process starts from position x with the mean where it starts from the jump-neighbor of However, we can still obtain Poincaré-type inequalities, but with a constant which is a polynomial of order higher than one. This power is higher than the constant , the optimal obtained in  for a path space of Poisson point processes.
It should be noted that the aforementioned translation property relates with the criterion (see [22,23]) for the Poincaré inequality, which states that if
then the Poincaré inequality is satisfied. A more detailed discussion on this criterion follows later in Section 2.2. Since this is not satisfied in our case we obtain a Poincaré-type inequality instead.
Before we present the results of the paper it is important to highlight an important distinction on the nature of the initial configuration from which the process can start. We can classify the initial configurations according to the return probability to them. Recall that the membrane potential of every neuron i takes positive values within a compact set and that whenever a neuron j different than i spikes, the neuron i jumps positions up, while the only other movement it does is jumping to zero when it spikes. That means that every variable can jump down only to zero while after the first jump, can only pass from a finite number of possible positions since the state-space is then discrete inside a compact due to the imposed maximum potential Since the neurons stay still between spikes, that implies that there is a finite number of possible configurations to which can return after every neuron has spiked for the first time. This is the domain of the invariant measure of the semigroup , and we will denote it as . Thus, if the initial configuration does not belong to , after the process enters , it will never return to this initial configuration.
It should be noted that it is easy to find initial configurations . For example one can consider any x such that at least one of the s is not a sum of synaptic weights , or any x with for every i and j.
Below we present the Poincaré inequality for the semigroup for general starting configurations.
Assume the PJMP as described in (2)–(5). Then, for every , the following Poincaré-type inequality holds.
with a second order polynomial of the time t that does not depend on the function f and β a constant.
One notices that since the coefficient is a polynomial of second order and is a constant, the first term dominates over the second for long time t, as shown in the next corollary.
Assume the PJMP as described in (2)–(5). Then, for every , and t sufficiently large, i.e.,
where is a constant depending only on
One should notice that although the lower value depends on the function f, the coefficient of the inequality does not depend on the function f.
The proof of the Poincaré inequality for the general initial configuration is presented in Section 2.
In the special case where x is on the domain of the invariant measure we obtain the stronger inequality.
Assume the PJMP as described in (2)–(5). Then, there exists a such that for every and for every , the following Poincaré-type inequality holds.
with a third order polynomial of the time t that do not depend on the function f.
As in the general case, for t large enough we have the following corollary.
Assume the PJMP as described in (2)–(5). Then, there exists a , such that for every , and t sufficiently large, i.e.,
where is a constant depending only on ,
We conclude this section with the Poincaré inequality for the invariant measure presented on the next theorem.
Assume the PJMP as described in (2)–(5). Then π satisfies a Poincaré inequality
for some constant .
2. Proof of the Poincaré for General Initial Configurations
In this section, we focus on neurons that start with values on any possible initial configuration as described by (2)–(5), and we prove the local Poincaré inequalities presented in Theorem 1 and Corollary 1. Let us first state some technical results.
2.1. Technical Results
We start by showing properties of the jump probabilities of the degenerate PJMP processes. Since the process is constant between jumps, the set of reachable positions y after a given time t for a trajectory starting from x is discrete. We therefore define
This set is finite for the following reasons. On one hand, for each neuron the set is discrete and such that the intersection with any compact is finite. On the other hand, we have
The idea is that since the process is constant between jumps, elements of are such that there exists a sequence of jumps leading from x to Since we are only interested on the arrival position among all jump sequences leading to we can consider only sequences with minimal number of jumps and the number of such jump sequences leading to positions inside a compact is finite, due to the fact that each is non-negative.
Since x is also in the compact we can have an upper bound for the cardinal of independent from
For a given time and a given position we denote by the probability that starting at time 0 from position the process has no jump in the interval and for a given neuron by the probability that the process has exactly one jump of neuron i and no jumps for other neurons.
Introducing the notation and given the dynamics of the model, we have that
As a function of is continuous, strictly increasing on and strictly decreasing on and we have
Assume the PJMP as described in (2)–(5).There exists positive constants and independent of and y such that
For all we have
For all we have
As said before, the set is finite so it is sufficient to obtain an upper bound for the ratio
We have for all
Here we decomposed the numerator according to two events. Either and there no jump in the interval of time or there is at least one jump in the interval of time whatever the position of the process at time
From the previous inequality, we then obtain
where we recall that the constant m appears in the definition of the compact set D introduced in (4).
Let us first assume that Recall that is defined in (7).
If we have
Assume now that and let us recall that as a function of is continuous, strictly increasing on and
On the other hand, as a function of is continuous and takes value for
we will need to control the ration . As shown in Lemma 1, this ratio depends on time when , otherwise it is bounded by a constant. For this reason, we will start by breaking the integration variable of the time t into two domains, and .
The first summand is upper bounded by the previous lemma. To bound the second term on the right-hand side of (13) we write
The distinction on the two cases, whether after time u the neurons configuration is or not, relates to the two different bounds Lemma 1 provides for the fraction whether y is or not. We will first calculate the second term on the right-hand side. For this term we will work similar to Lemma 2. At first we will apply the Holder inequality on the time integral after we first divide with the normalization constant . This will give
Now we will use the Cauchy-Schwarz inequality in the first sum. We will then obtain the following
The first term on the last product can be upper bounded from Lemma 1
Meanwhile for the second term involved in the product of (15) we can write
where for the first bound we made use once more of the Holder inequality, after we divided with the appropriate normalization constant . If we put the last bound together with (16) into (15), we obtain
We now calculate the first summand of (14). Notice that in this case we cannot use the analogue bound from Lemma 1, that is as we did for , since that will lead to a final upper bound which may diverge. Instead, we will bound the by the carré du champ of the function after the first jump. We can write
where above we divided with the normalization constant , since . We can now apply the Holder inequality on the sum, so that
If we combine this together with (17) and (14) we get the following bound for the second term of (13)
The last one together with the bound shown in Lemma 2 for the first term of (13) gives
since the carré du champ is non-negative, as shown below
by Cauchy-Swartz inequality. □
We have obtained all the technical results that we need to show the Poincaré inequality for the semigroup for general initial configurations.
2.2. Proof of Theorem 1
Denote . Then
We can write
If we could use the translation property used for instance in proving Poincaré and modified log-Sobolev inequalities in [13,17], then we could bound relatively easy the carré du champ of the expectation of the functions by the carré du champ of the functions themselves, as demonstrated below
The inequality for relates directly with the criterion (see [22,23]) which states that if then the Poincaré inequality is true, since
Unfortunately, this is not the case with our PJMP where the degeneracy of jumps and the memoryless nature of them allows any neuron to jump to zero from any position, with a probability that depends on the current configuration of the neurons. Moreover, contrary on the case of Poisson processes, our intensity also depends on the position.
To obtain the carré du champ of the functions we will make use of the Dynkin’s formula which will allow us to bound the expectation of a function with the expectation of the infinitesimal generator of the function which is comparable to the desired carré du champ of the function.
Therefore, from Dynkin’s formula
To bound the second term above we will use the bound shown in Lemma 3
For the second term we can bound by M. For the last term on the right-hand side, since the carré du champ is non-negative, we can get
where above we used the property of the Markov semigroup . Since this last quantity does not depend on s, and we further get
Putting everything together we finally obtain
And so, the theorem follows for constants
3. Proof of the Poincaré Inequalities for Starting Configuration on the Domain of the Invariant Measure
We start by showing that in the case where the initial configuration belongs on the domain of the invariant measure, we obtain a strong lower bound for the probabilities , for time t big enough, as presented on the following lemma.
Assume the PJMP as described in (2)–(5). Then, for every and
Since is finite, we know that there exists a strictly positive constant , such that for every . Since, for every , such that , we obtain that there exists a such that for every , such that , since is finite. □
Taking under account the last result, we can obtain the first technical bound needed in the proof of the local Poincaré inequality, taking advantage of the bounds shown for times bigger than .
Assume the PJMP as described in (2)–(5). Then, for every
for every .
We can compute
Now we will use three times the Cauchy-Schwarz inequality, to pass the square inside the integral and the two sums. We will then obtain
where above we used the semigroup property . Since , we can use Lemma 4 to bound for every w and , . We then obtain
Proof of Theorem 2
We can now show the Poincaré inequality for the semigroup for initial configurations inside the domain of the invariant measure .
We will work as in the proof of the Poincaré inequality of Theorem 1 for general initial conditions. As before, we denote . Then
since . To bound the carré du champ, as in the general case of Theorem 1, from (19) and Dynkin’s formula we obtain the following
Since , we can use Lemma 5 to bound the second term on the right-hand side
4. Proof of the Poincaré Inequalities for the Invariant Measure
In this section, we prove a Poincaré inequality for the invariant measure presented in Theorem 3, using methods developed in [20,24,25].
At first assume . We can write
We will follow the method from  used to prove Spectral Gap for finite Markov chains. Since , we can write
Consider the shortest path from to , where the indexes stand for the neuron that spikes. Since is finite is finite. We denote and , for , so that . So, we can write
where above we used that . We can now form the caré du champ
We then have
Putting all together leads to
In this paper, we study the probabilistic model introduced by Galves and Löcherbach in , in order to describe neural networks. In particular, we want to show that despite the degenerate nature of the process, one can still obtain Poincaré-type inequalities for the associated semigroup, similar to the ones obtained by [13,14,17] for point processes without degenerate jumps.
In terms of practical applications, the concentration inequality we have derived for the invariant measure , implies that the process remains very close to its mean, while the Poincaré-type inequalities for the semigroup, imply that despite the degeneracy that characterizes the behavior of the neural system, after a long time passes the neurons behave closer to a system without degeneracy.
In the current paper we studied both the cases where the initial configuration, belongs and does not belong in the domain of the invariant measure. Future directions should focus on restricting to the special case where the initial configuration belongs exclusively in the domain of the invariant measure. Then, stronger inequalities from the Poincaré-type inequalities obtained here for the semigroup appear to be satisfied, as is the modified logarithmic Sobolev inequality.
Furthermore, the extension of the results obtained in the current paper for compact neurons, to the more general case of unbounded neurons is of particular interest.
The authors contributed equally to this work.
This article was produced as part of the activities of FAPESP Research, Innovation and Dissemination Center for Neuromathematics (grant 2013/ 07699-0, S.Paulo Research Foundation); This article is supported by FAPESP grant (2016/17655-8) and (2017/15587-8) .
The authors thank Eva Löcherbach for careful reading and valuable comments.
Conflicts of Interest
The authors declare no conflict of interest.
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