A Density Model for a Population of Theta Neurons
© Dumont et al.; licensee Springer. 2014
Received: 15 January 2013
Accepted: 3 September 2013
Published: 17 April 2014
Population density models that are used to describe the evolution of neural populations in a phase space are closely related to the single neuron model that describes the individual trajectories of the neurons of the population and which give in particular the phase-space where the computations are made. Based on a transformation of the quadratic integrate and fire single neuron model, the so-called theta-neuron model is obtained and we shall introduce in this paper a corresponding population density model for it. Existence and uniqueness of a solution will be proved and some numerical simulations are presented. The results of existence are compared to previous results of existence or nonexistence (burst) for populations of leaky integrate and fire neurons.
It is a big challenge to find the most appropriate mathematical model to describe the electrical activity of populations of neurons; it should, in the first place, give a realistic view of the very complex brain activity and be able to describe the emergent phenomena that are observed in vivo, but in the same time, it should keep a certain simplicity that would help to analytically solve it and to numerically implement it.
Our attention has been kept by the so-called population density approach that has been successfully used to describe the evolution of physiologically structured population in many areas of biology, and in particular, in neuroscience. A population density model will track down the evolution of a density function of the population in the state space, which is determined by the structuring variable. In theoretical neuroscience, the concept of a probability density function has been already extensively used ([1–3]). A step forward, though, was made by applying this concept to model interactions of large populations of sparsely connected neurons ([4–7]). The connection between probability-density approach and population density approach is based on the observation that, for a large population of similar neurons, the probability density can be interpreted as a population density ([4, 6]). For a method to derive population density models, we refer to , where an illustrative exemplification is given for the case of integrate-and-fire neurons. Here, the effect of the synaptic connections has been modeled as a jump in the state variable, the membrane potential in this case, when a neuron of the population receives a synaptic input. For more simulations of networks of integrate-and-fire neurons via population density models, we also refer to  and . Another method can be found in  where a population density equation has been derived for a population of SRM (spike-response model) neurons with escape noise. A well-posedness result for a population density model of Leaky Integrate-and-Fire (LIF) neurons can be found in . The approach proved to be an useful tool in analyzing special behaviors of neural populations, such as the existence of equilibrium solution (), or the emergence of synchronization of neurons ([13–15]).
It is somehow usual to apply the population density formalism to populations of integrate-and-fire neurons, due to the simplicity of the model and to the possibility to express the firing rate in terms of the population density function. We have chosen in this paper to consider a large homogeneous population of neurons that are characterized by the theta-neuron model (). As it is known, the theta neuron model, or Ermentrout–Kopell model, is an alternative version of the Quadratic Integrate-and-Fire (QIF), which is the simplest spiking neuron model. In contrast to the leaky integrate-and-fire model, the QIF model does have a spike generation mechanism, which makes it suitable for us to describe the internal state of a population density function of neurons. Nevertheless, the use of the equivalent theta-neuron model is preferable since it is a continuous version of the QIF model, and the state variable varies in a finite domain. We will come back in the first section of this paper with more details about this subject.
We therefore use the population density formalism in this paper to derive a population density model for a population of theta-neurons and we shall prove the well-posedness of the model by a method similar to those used in  or  in the case of populations of leaky integrate-and-fire neurons. The main difference between these cases and the one considered in this paper is due to the different expressions of the firing rates of the populations.
The paper is structured as follows: In the first section, the method used in  to obtain a population density model for integrate-and-fire neurons is adapted to the case of a homogeneous population of neurons characterized by the quadratic integrate-and-fire model. Based on the Ermentrout–Kopell transformation, the quadratic integrate-and-fire can be written in its equivalent form in terms of a new variable called the phase of a neuron. We next introduce a population density model for the population of neurons that is structured by their phase instead of their membrane’s potential. We continue by proving the well-posedness of the model; in the non-connected case, i.e., when all the neurons of the population receive only an external stimulus, the result we prove is global. In the case of a connected population, we prove a global well-posedness result under an assumption that has sense from a biological point of view. If the above specified assumption is not taken into consideration, the result is only local.
We end this paper by presenting some numerical simulations for the population density model that we introduced, which are compared to direct Monte Carlo simulations.
2 Quadratic Integrate-and-Fire Neurons: Population Density Approach
Here, represents the potential of the neural membrane at time t, are the arrival times of external impulses, and the effect of the reception of a spike at neuron’s synapse has been modeled as a jump of size h of the potential v. The jump is positive (respectively negative) if the spike is received from an excitatory (respectively inhibitory) source. Due to the quadratic term, v can reach infinity in finite time. The time when v reaches the infinity value is considered as the time when the neuron is emitting a spike and the potential of the membrane is instantaneously reset to −∞. The parameter plays a key role in the dynamics of the (QIF) model of neuron’s potential (see [18, 19], and  for details).
One of the main hypotheses used to obtain a population density model is that the population is homogeneous, i.e., all the neurons of the population have the same properties, and, in our case, are individually described by the model (1).
In the model above, the case of instantaneous reception of the impulses can be obtained by taking .
Note that if the initial condition satisfies , then the solution to the nonlinear problem (10) also satisfies .
In our paper, stands for the rate of the Poisson spike train that each neuron receives from an external source, which is not explicitly modeled. The rate is then considered as given. The case of a probability density model where the Poisson spike train is approximated by the sum of a deterministic baseline and a white noise has been considered in . In the paper , the authors derived an explicit formula of the firing rate of a noisy quadratic integrate-and-fire neuron with and without the synaptic dynamics. It is possible to look at this formula as the second-order approximation of the firing rate of a neural network where each neuron receives an independent Poisson spike train.
In the paper , the authors study the firing rate of the noisy quadratic integrate-and-fire neuron receiving an oscillatory input. To this end, the authors used the so-called linear response theory. The theory is not really adapted to a neural network where each neuron receives an independent Poisson spike train since the transfer function cannot be computed explicitly.
3 A Population Density Model for Theta Neurons
That means that the reset mechanism in (11) is replaced in this model by the simple passing of the phase of the neurons, θ, through the value 2π.
Based on the transformation of the model (1) into the model (17), we intend to obtain a corresponding population density model for a population of neurons characterized by their phase θ. The advantages of doing so are obvious: first of all, through this transformation, the state space is transformed into a finite one . More than that, the reset mechanism which creates a discontinuity in the state v will be replaced by a continuous flow through the state 2π, which will influence the expression of the firing rate of the population, as it can be seen below.
where the functions f and s are defined by (19).
We can underline now few differences between the expression of the firing rate in the case of a theta-neuron population and that of a population of leaky integrate-and-fire neurons. The first one has been stated above; if in the case of leaky integrate-and-fire populations, the firing rate was taking into account only the “jumping” part of the flux, we have here the opposite case, since only the drift flux influences the rate of neurons at the firing phase. Another major difference is that, in our model, the firing rate does not explicitly depend on the average reception rate σ as it is the case in the leaky integrate-and-fire population density models ([6, 11]).
Using the boundary condition, and integrating (20) on the domain , one can easily check the conservation property of Eq. (20).
where, as before, if we take α as a given function of time, we obtain the case where synaptic delays are considered, whereas for , we obtain the case of instantaneous synaptic transmission.
4 Existence and Uniqueness of the Solution
In this section, we shall prove the existence and uniqueness of the solution to problem (22). This will be done first in the linear case, i.e., when (with a given function), and later in the general nonlinear case.
4.1 The Linear Case
where is a given continuous function. The main result of the subsection is stated below.
We have used here a different notation for a curve starting from a point in order to avoid confusions. Due to the properties of the function f, we will have that, for any given point there is a unique initial point .
therefore we have that for any finite t, is strictly positive, and then the characteristics starting from different points do not cross. Nevertheless, depending on the sign of the above derivative can go asymptotically to 0.
which ends the proof.
4.2 The Nonlinear Case
Let us go back now to the general model (22). Below, we prove the existence and uniqueness of a solution locally in time. Then, under an assumption regarding the number of connections per neuron and the delay kernel, the global in time existence is proved.
Theorem 2 Let and α be two functions of and the initial condition be a periodic function of . Then one can find such that there exists a unique positive solution to the nonlinear problem (22), , which is periodic with respect to the second argument.
The proof will use the standard Banach–Picard fixed-point theorem applied to the map G with respect to the usual norm on . Below will denote the norm in .
has a fixed point.
where we have also used the fact that .
which shows that the invariance of the ball property takes place locally in time.
one gets the conclusion on the interval .
Theorem 3 Let us assume the same hypothesis as in Theorem 2. We assume furthermore that . Then there exists an unique solution to problem (22) that is global in time.
with c a positive constant to be specified later. By doing so, we obtain the result on the interval of length , and since the harmonic series is divergent, by making , we will get the existence and uniqueness of the solution on . It remains to show that the inequality (40) holds.
which completes our proof.
and q is already calculated on the interval , one gets a global solution for this special case, which is given by the solution of the linear problem.
5 Numerical Results
where stands for the integral part of the flux and for the drift part of the flux.
The drift numerical flux was reconstructed by using the upwind method (see  for details of the upwind numerical reconstruction) and the integral part was approximated by using a first order reconstruction.
6 Discussion and Conclusion
Single neuron models such as the LIF or the QIF models have a weak electrophysiological basis, but thanks to their simplicity, they are quite useful for simulations of the behavior of populations of neurons. The population density approach leading to partial differential equations is suited for very large populations of neurons; we think that mathematical studies on the qualitative behavior of population density models may help for the choice of the particular single neuron model used to describe the internal state of the neurons of the population, and give insights on the results. In particular, the possibility of burst of the firing rate corresponding to a synchronization of the neurons, as opposed to a regular activity, is of interest to neuroscientists.
We have highlighted a qualitative difference between the population density approach applied to a population of theta-neurons, and the same approach applied to populations of LIF neurons. In , it was proved that a global solution exists for the LIF population density approach equation with no delay and the firing rate remains bounded in the case where . On the other hand, in  it was shown that for J and large enough, for any initial condition there will be a burst in finite time: The firing rate goes to infinity.
In the present study of populations of theta-neurons, we consider only the case with conduction delay. The condition for existence of a global solution (no burst) involves the product of the number of connections J and the maximum of the delay repartition . In order to satisfy this condition for large J, the delay kernel α should spread over the time interval in order to decrease its maximum. So, it is possible to exhibit populations with the same J that will burst in finite time with the LIF model but will have a regular behavior on an infinite horizon with the QIF model with a different delay repartition.
As it is known, the formal threshold imposed in the LIF model is defined as the value at which an action potential is initiated, and the firing rate of the population density models in this case is defined as the flux passing through this threshold. In our case, the neurons of the population are supposed to transmit the electrical signal at the peak value, instead of the value at which the initiation of a spike occurs, which is actually the root of the model (19). Therefore, the firing rate in our model depends only on the drifting flux through the phase 2π. This fact allowed us to obtain a global well-posedness result in the general case of the model. But the same fact does not allow to use the same argument as in  to study the conditions of bursting. Furthermore, in all the simulations that we have done, the synchronization phenomenon have not been observed in the case of a theta-neurons population.
The second and third authors were members of the project LEA Math-Mode Projet Franco-Roumain. The first author has been financially supported by Conseil Régional d’Aquitaine.
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