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

For

Then, we restrict ourselves to the special case where the initial configuration

Then we show a Poincaré inequality for the invariant measure

Before we describe the model, we present the neuroscience framework of the problem.

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

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 [

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 [

We consider a process

Working with Hawkes processes allows consideration of systems with infinitely many neurons, as in [

Let

In the above equation for each

This can be seen in the following way. The first term

The generator of the process

Furthermore, we also assume the following conditions about the intensity function:

The probability for a neuron to spike grows with its membrane potential so it is natural to think of the function

Our purpose is to show Poincaré-type inequalities for our PJMP, whose dynamic is similar to the model introduced in [

We will investigate Poincaré-type inequalities at first for the semigroup

Let us first describe the general framework and define the Poincaré inequalities on a discrete setting (see also [

We consider a Markov process

We define

Furthermore, we define the “carré du champ” operator (see [

For more details on this important operator and the inequalities that relate to it one can look at [

We say that a measure

For the Poincaré inequality for continuous time Markov chains one can look in [

The main difficulty here is that for the pure jump Markov process that we examine in the current paper, the translation property

It should be noted that the aforementioned translation property relates with the

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

It should be noted that it is easy to find initial configurations

Below we present the Poincaré inequality for the semigroup

One notices that since the coefficient

One should notice that although the lower value

The proof of the Poincaré inequality for the general initial configuration is presented in

In the special case where

As in the general case, for

We conclude this section with the Poincaré inequality for the invariant measure

In this section, we focus on neurons that start with values on any possible initial configuration

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

This set is finite for the following reasons. On one hand, for each neuron

The idea is that since the process is constant between jumps, elements of

Since

For a given time

Introducing the notation

Define

As a function of

As said before, the set

We have for all

Here we decomposed the numerator according to two events. Either

From the previous inequality, we then obtain

Let us first assume that

If

Assume now that

On the other hand, as a function of

We deduce from this that there exists

Now (

For all

Using the explicit value of

Recall that

In both cases, when

From this, we deduce that there exists a constant

Putting all together, we obtain the announced result for the case where

We now consider the case where

We start by considering the case where

As a function of

We deduce from this that there exists

Let us assume now that

Recall the explicit expression of

Taking under account the last result, we can obtain the first technical bound needed in the proof of the local Poincaré inequality.

Consider

To continue we will use Holder’s inequality to pass the second power inside the first integral, which will give

The first quantity involved in the above integral is bounded from Lemma 1 by a constant

We will now extend the last bound to an integral on a time domain starting at

To calculate a bound for

The first summand

The distinction on the two cases, whether after time

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 (

We now calculate the first summand of (

If we combine this together with (

The last one together with the bound shown in Lemma 2 for the first term

We have obtained all the technical results that we need to show the Poincaré inequality for the semigroup

Denote

We can write

If we could use the translation property

The inequality

Unfortunately, this is not the case with our PJMP where the degeneracy of jumps and the memoryless nature of them allows any neuron

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

This together with (

Finally, plugging this in (

For the second term we can bound

Putting everything together we finally obtain

And so, the theorem follows for constants

We start by showing that in the case where the initial configuration belongs on the domain

Since

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

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

We can now show the Poincaré inequality for the semigroup

We will work as in the proof of the Poincaré inequality of Theorem 1 for general initial conditions. As before, we denote

From that and (

Since

In this section, we prove a Poincaré inequality for the invariant measure

At first assume

We will follow the method from [

Consider

We then have

Putting all together leads to

In this paper, we study the probabilistic model introduced by Galves and Löcherbach in [

In terms of practical applications, the concentration inequality we have derived for the invariant measure

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)

The authors thank Eva Löcherbach for careful reading and valuable comments.

The authors declare no conflict of interest.