Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation Equation
 Khashayar Pakdaman^{1}Email author,
 Benoît Perthame^{2, 3, 4} and
 Delphine Salort^{1}
https://doi.org/10.1186/21908567414
© K. Pakdaman et al.; licensee Springer 2014
Received: 15 December 2012
Accepted: 18 June 2013
Published: 24 July 2014
Abstract
Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge.
In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two “extreme” cases.
Mathematics Subject Classification (2000)2010:35B40, 35F20, 35R09, 92B20.
Keywords
1 Introduction
This article is devoted to study the large time behavior of the solution to a conservative aggregationfragmentation equation, a class of equations that arises in many applications and that has been widely studied both in the linear case [8, 14, 18, 19] and with nonlinearities [6, 9–11, 13, 20].
Our particular motivation is an extension of the elapsed time neural population model, a partial differential equation structured by “age” studied in [15–17], and which gives a new approach to the understanding of synchronization/desynchronization of neural assemblies with respect to the strength of their interconnections. Here, we add a fragmentation term in the model in order to incorporate the fact that the dynamics of the neurons are also related to their past activity, notably that neurons display adaptation and fatigue. That is a progressive decrease of their propensity to firing in response to a step maintained current. This is one of the most common neuronal properties that can introduce correlation in firing times. In this work, we examine whether and how the inclusion of this property can affect the dynamics of neural assemblies. As a consequence, the mathematical study of this equation is more complex. Based on the ideas in [12], we give a new result of exponential decay of the solution to its stationary state in the case where the network is weakly connected.

denotes the probability density of neurons at time t such that the time elapsed since the last discharge is s. It is a fundamental property which follows from our assumptions that, for all times $t\ge 0$,$n(s,t)$${\int}_{0}^{+\mathrm{\infty}}n(s,t)\phantom{\rule{0.2em}{0ex}}ds={\int}_{0}^{+\mathrm{\infty}}{n}^{0}(s)\phantom{\rule{0.2em}{0ex}}ds=1,\phantom{\rule{1em}{0ex}}n(s,t)\ge 0.$(2)

represents the flux of neurons which discharge at time t and is identified to the global amplitude of stimulation of the network.$N(t)$

models the firing rate of neurons submitted to a stimulation of amplitude N and such that the time elapsed since the last discharge is s. The coupling between the neurons is taking into account via the function p which varies according to the global activity $N(t)$. Hence, in this model, the strength of interconnections between the neurons is taking into account via the variations of p with respect to the variable N.$p(s,N)$

The kernel $K(s,u)\in \mathcal{M}([0,\mathrm{\infty})\times [0,\mathrm{\infty}))$, the set of nonnegative measures in $[0,\mathrm{\infty})\times [0,\mathrm{\infty})$, gives the distribution of neurons which take the state s when a discharge occurs after an elapsed time u since their last discharge.
The structured nature of Eq. (1) is related to the choice of the description of the dynamic of the neurons, which is made via the time elapsed since their last discharge. The term “fragmentation” stems from the fact that, at each time, the density of neurons which discharge is fragmented, via K, with respect to the new state of neurons after their discharge; each fragment is given by the flux of neurons which discharge and come back in a same state s.
The existence of a stationary solution is proved in Sect. 5 and we attack to convergence through an adaptation of the strategy in [16]. For the linear problem, we construct some kind of spectral gap which opens the door to also treat ‘small’ (in a weak sense) nonlinearities.
The paper is organized as follows. In Sect. 2, we state our main results after giving assumptions on the coefficients; we separate the linear and nonlinear cases because we can prove much stronger results in the linear case. In Sect. 3, we study the solution of the linear version of Eq. (1) more precisely we prove its large time convergence to the stationary state with exponential decay; this is the proof of Theorem 2.1. Section 4 is devoted to the nonlinear case and to the proofs of Theorems 2.2 and 2.3. We prove the existence of stationary states, i.e., solutions to (3) in Sect. 5. In the last Sect. 6, we present numerical results in the case where the nonlinearity is strong enough to obtain periodic solutions to understand the effect of the fragmentation term in regard to the appearance of spontaneous activity in the network. Several general or technical results are postponed to Appendices in order to focus more the proofs on the main arguments.
2 Assumptions and Main Results
We need technical assumptions on the coefficients p and K in (1) and before we write them in full generality, we begin with a particular example. For the kernel K, a Dirac mass at 0, $K(s,u):={\delta}_{s=0}$, the equation is equivalently written as a age structured equation and this situation is covered in [16, 17]. In this case, the interpretation is clear: After they discharge, all neurons take the same state $s=0$, irrespective of the time elapsed since their last discharge.
where ψ is a given increasing function. In this situation, the post discharge state s of the neurons only depends on their discharge state u (a cumulative time elapsed since their last discharge). Still more general is when $K(s,\cdot )$ is a function: this includes variability in the neuron population or randomness in their behavior.
with σ a given smooth function. It is a caricature for modeling three desirable properties of the neurons:

immediately after a discharge, the neuron enters a refractory period, i.e., after a discharge, a neuron cannot discharge again during a certain time interval; this is the assumption that $p(s,N)=0$ for s small,

after the end of its refractory period, the neuron rapidly recovers a significative sensitive state,

for an excitatory system, a larger stimulation on the neuron induces smaller refractory period that $\frac{\partial}{\partial N}p(s,N)>0$ and this is written here as ${\sigma}^{\prime}<0$ even though this assumption is not used in the present analysis.
Those examples of functions p and K are covered by more general assumptions, and links between these quantities, which we explain now. In Appendix A, we give explicitly the conditions on the two functions σ and ψ which are induced by our assumptions below.
2.1 General Assumptions
Assumptions on the Rate $p(s,N)$
In other words, the nonlinearity in (1) is well determined.
Assumptions on the Distribution $K(s,u)$
These assumptions are fundamental in order to guarantee that $n(\cdot ,t)$ is a probability as written in (2), but also that $\int K(s,u)p(u)n(u)\phantom{\rule{0.2em}{0ex}}du$ is well defined for n an integrable function.
Assumptions Linking p and K
The following assumptions which link p and K allow us to prove, in the case of weakly connected neurons, convergence of the solution of Eq. (1) to the stationary state $A(s)$ in (3) with an exponential rate.
This is again to say that σ is small enough but not necessarily ${p}_{M}$.
2.2 Exponential Decay for the Linear Equation
Our first theorem gives a result of exponential decay of solutions to the linear equation of (1) toward the steady state A built in Sect. 5. There are several routes toward this goal. A spectral gap can be proved using Poincaré type inequalities; this idea has been developed in [1, 4, 5] and is for smooth kernels K. A probabilistic approach has also been developed; see [2, 3] and the references therein.
With these notations and the function $P(s)>0$ (with ${p}_{\ast}=p$ in the linear case at hand) constructed in Appendix C, we can state our first result.
Theorem 2.1 (Exponential decay. Linear case)
2.3 Exponential Decay for the Nonlinear Equation
With the notations and preparation of the linear case, we can state our second theorem on exponential decay for weakly nonlinear equation and when p is smooth enough. For a better presentation of the proofs, we separate our statements in two theorems.
Theorem 2.2 (Exponential decay. Nonlinear case)
Theorem 2.3 (Decay on m)
3 Exponential Decay for the Linear Equation
The strategy of the proof of Theorem 2.1 is to observe that exponential decay for $n(s,t)A(s)$ follows from exponential decay for the functions $M(s,t)$ and its first time derivative, which is much easier to prove than for m itself; the counterpart is that it involves a weighted norm, as expressed in Theorem 2.1, at variance with the Poincaré method in [1, 4, 5]. Indeed, there are two main advantages of considering the solutions $M(s,t)$, $J(s,t)$, instead of $m(s,t):=n(s,t)A(s)$; (i) they satisfy a closed equation, (ii) the dual problem to the corresponding stationary equation has a negative first eigenvalue. This directly implies exponential decay of both ${\int}_{0}^{+\mathrm{\infty}}P(s)M(s,t)\phantom{\rule{0.2em}{0ex}}ds$ and ${\int}_{0}^{+\mathrm{\infty}}P(s)J(s,t)\phantom{\rule{0.2em}{0ex}}ds$.
We split the proof of Theorem 2.1 in two steps:

In the first part, we check that the proof of Theorem 2.1 is a direct consequence of the exponential decay in ${L}^{1}$ of $P(s)M(s,t)$ and $P(s)J(s,t)$.

The second part is devoted to prove exponential decay in ${L}^{1}$ of $P(s)M(s,t)$ and $P(s)J(s,t)$.
3.1 Reduction to Exponential Decay on $M(s,t)$ and $\frac{\partial}{\partial t}M(s,t)$
We derive the Theorem 2.1 from the following proposition.
Next, we control the small values of s, namely ${\int}_{0}^{\sigma}m(s,t)\phantom{\rule{0.2em}{0ex}}ds$. We cannot control this quantity directly and proceed to estimate ${\int}_{0}^{\sigma}{e}^{\mu s}m(s,t)\phantom{\rule{0.2em}{0ex}}ds$ for μ given in condition (16).
This concludes the proof of Proposition 3.1. □
3.2 Exponential Decay of M and $\frac{\partial}{\partial t}M$
We now establish the assumptions in Proposition 3.1 on exponential decay for M and J. This is stated in the following proposition.
assuming the initial bounds (20) are satisfied.
Proof We divide the proof in two steps. We first derive a closed form for the equation on $M(s,t)$; this is our main observation, which allows to extend the argument in [12] to nonconstant coefficients. Then, thanks to dual problem that we study in Appendix C, we conclude our proof.
There are to routes to go further. To cover the case of interest where p can vanish during the refractory period, we use a duality argument. However, we can use directly formula (30) under different assumptions on p; this is performed in Appendix B.
which proves the decay of $M$ in Proposition 3.2 thanks to the Gronwall lemma.
which concludes the proof of Proposition 3.2. □
4 Exponential Decay for the Nonlinear Case
The proofs of Theorems 2.2 and 2.3 follow the strategy used to prove Theorem 2.1. The main difficulty is that the control on M is a weak control on m while the nonlinear term, which involves $N(t)$, is in a strong dependency. To solve this difficulty, we had to assume that the function p is regular enough; this allows us, via an integration by parts to increase the regularity of the nonlinear term $N(t)$ at the expense of Lipschitz regularity on p.
which proves the inequality (21) using the Gronwall lemma.
Inserting this exponential decay on $\int P(s)M(s,t)\phantom{\rule{0.2em}{0ex}}ds$ in (35) proves (22).
Step 2. Proof of ( 23 ). In order to better explain our strategy, we begin with a global Lipschitz estimate on N and then prove the exponential decay.
The Lipschitz bound on p and (2) gives the estimate ${N}^{\prime}\le C$.
All these terms have exponential decay, and thus, back to inequality (36), this concludes the proof of (23).
The proof of Theorem 2.2 is now complete. □
thanks to the exponential decay on ${N}^{\prime}$ in (23).
Putting the estimates (39) and (40) together, we conclude the proof of estimate (38).
Using the same computations than in the proof of Proposition 3.1 and using the bound (33) on R, we obtain Theorem 2.3. □
5 Existence of a Stationary State
This section is devoted to the proof of existence and uniqueness of stationary states for Eq. (1) in the case where the network is weakly connected. We begin with the linear case and then we treat the weakly nonlinear case that is weakly connected networks.
5.1 The Linear Case
The following theorem holds.
Theorem 5.1 (Stationary states. Linear case)
The proof is standard [8, 14, 18, 19] and based on the Krein–Rutman theorem (see [7]).
Proof of Theorem 5.1 To justify the computations in this proof, we restrict ourselves to the case when p and K are continuous, which allows us to use a consequence of the Krein–Rutman theorem as recalled in Theorem D.1; this is not a restriction because the extension to our assumptions in Sect. 1 follows from standard regularization argument and passing to the limit as we do it below.
To prove Theorem 5.1, we need to pass to the limit in Eq. (43) when ε and ${R}^{1}$ go to 0. To do this, it is enough to prove compactness for the eigenvalues ${\lambda}_{\epsilon ,R}$ and convergence for the functions ${A}_{\epsilon ,R}$ to a function A satisfying properties of Theorem 5.1.
We begin with the following a priori estimates
where θ is defined in (12) and ${s}^{\ast}$, ${p}^{\ast}$ are defined in (41).
The formula (46) concludes the lower estimate for ${\lambda}_{\epsilon ,R}$ and the proof of the Lemma 5.2. □
We continue our a priori estimates with the following lemma.
and, from the estimate in Lemma 5.2, we deduce the ${L}^{\mathrm{\infty}}$ bound on ${A}_{\epsilon ,R}$ in Lemma 5.3. The other bound follows directly from the equation. □
To conclude the existence proof of Theorem 5.1, we pass to the limit in the weak form of (43) as $\epsilon \to 0$, $R\to \mathrm{\infty}$.
Uniqueness is a standard property in Krein–Rutman theory and we refer to [18] for the particular example at hand. □
5.2 The Nonlinear Case
Theorem 5.4 Assume (4)–(8), (12). Then there is a steady solution $A(s)$ to (1).
Therefore, there is at least one steady state. □
Notice that uniqueness is expected with the smallness assumptions (7) because ${F}^{\prime}(N)$ should be small (here, we leave this point without proof).
6 Numerical Simulations and Spontaneous Activity
When our smallness assumptions is not fulfilled, the neurons may undergo synchronization leading to a spontaneous activity of the network. The aim of this section is to illustrate this regime through numerical simulations and show the effect of the fragmentation term when the flux of neurons $N(t)$ does not converge to a stationary state but oscillates.
To do so, we compare the dynamic of N in the two following “extreme” cases

which is the case studied in [17] where all the neurons come back in a same state after discharge,$K(s,u)={\delta}_{s=0}$

when the neurons, after discharge, reach a state which is proportional to their time elapsed since discharge.$K(s,u)={\delta}_{s=u/2}$
We now compare numerical simulations of the dynamic of N with the two kernels mentioned above.

For $K={\delta}_{s=0}$, with this choice of function p given by (47), theoretical study and numerical results in [17] have shown that for all $\alpha >0$, there exists a very large class of periodic solutions, Fig. 1 (left) depicts such a periodic solution. Moreover, with the numerical observations, the dynamic strongly depends on the initial data (see Fig. 2 and article [17]).

The kernel $K(s,u)={\delta}_{s=u/2}$ seems to create a “smoothing effect.” Indeed, unlike the former case, when α is small enough, the numerical solution N converges to a stationary state (see Fig. 1, right). Moreover, for α fixed large enough, it seems that there does not exist a large spectrum of periodic solutions as in the case where $K(s,u)={\delta}_{s=0}$; more precisely, we numerically obtain only one periodic solution (see Figs. 3 and 4). The numerical methods used here are analogous of those in article [17]; hence, we refer to this article for a description of the algorithm.
Appendix A: An Example of Coefficients p and K
Finally the conditions (15) and (16) are reduced to saying that ${\sigma}_{+}$ and θ are small enough.
Appendix B: A Direct Use of the Fundamental Formula on $M$
We can give a direct proof of exponential decay for M based on this expression and using simpler assumptions on p and K that, however, do not cover the case when p can have large variation and $p(s)$ can vanish for $s\simeq 0$. It is a natural extension of the case when p is constant as covered in [12].
We are going to prove the following proposition.
□
Appendix C: A Noncompact Eigenproblem
Its solution uses in a fundamental way the smallness assumptions of Sect. 2.1 linking p and K. Indeed, the following lemma holds.
Proof Equation (51) is a delay differential equation, and thus has a global solution, i.e., for $s\ge 0$, for all λ. We consider a time interval $[0,{s}_{0})$ where the solution P is positive and we prove that for $\lambda >0$ close enough to zero we can take ${s}_{0}=\mathrm{\infty}$. We argue for $\lambda =0$ and then by continuity for λ close enough to 0.
thanks to assumption (15). In particular, we conclude that ${s}_{0}>{\sigma}^{\ast}$.
Therefore, we obtain that for $\lambda <0$ close enough to 0 then the bracket is positive and P is increasing on $[{\sigma}^{\ast},+\mathrm{\infty})$ which proves Lemma C.1. □
Appendix D: A Consequence of the Krein–Rutman Theorem
To prove the existence of steady states, we have used the existence of a solution to a regularized eigenfunction problem. Namely, we have the
Declarations
Authors’ Affiliations
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