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The authors have declared that no competing interests exist.

Conceived and designed the experiments: MNG JT. Performed the experiments: MNG. Analyzed the data: MNG JT. Contributed reagents/materials/analysis tools: MNG JT. Wrote the paper: MNG JT.

Deriving tractable reduced equations of biological neural networks capturing the macroscopic dynamics of sub-populations of neurons has been a longstanding problem in computational neuroscience. In this paper, we propose a reduction of large-scale multi-population stochastic networks based on the mean-field theory. We derive, for a wide class of spiking neuron models, a system of differential equations of the type of the usual Wilson-Cowan systems describing the macroscopic activity of populations, under the assumption that synaptic integration is linear with random coefficients. Our reduction involves one unknown function, the effective non-linearity of the network of populations, which can be analytically determined in simple cases, and numerically computed in general. This function depends on the underlying properties of the cells, and in particular the noise level. Appropriate parameters and functions involved in the reduction are given for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models. Simulations of the reduced model show a precise agreement with the macroscopic dynamics of the networks for the first two models.

The activity of the brain is characterized by large-scale macroscopic states resulting from the structured interaction of a very large number of neurons. These macroscopic states correspond to signals experimentally measured through usual recording techniques such as extracellular electrodes, optical imaging, electro- or magneto- encephalography and magnetic resonance imaging. All these experimental imaging protocols indeed record the activity of large scale neuronal areas involving thousands to millions of cells. At the cellular level, neurons composing these columns manifest highly complex, excitable behaviors characterized by the intense presence of noise. Several relevant brain states and functions rely on the coordinated behaviors of large neural assemblies, and resulting collective phenomena recently raised the interest of physiologists and computational neuroscientists, among which we shall cite the rapid complex answers to specific stimuli

This motivates the development of models of the collective dynamics of neuronal populations, that are simple enough to be mathematically analyzed or efficiently simulated. A particularly important problem would be to derive tractable macroscopic limits of the widely accepted and accurate Hodgkin-Huxley model

The question of the macroscopic modeling of cortical activity and their relationship with microscopic (cellular) behavior has been the subject of extensive work. Most studies rely on heuristic models (or firing-rate models) since the seminal works of Wilson, Cowan and Amari

Despite these efforts, deriving the equations of macroscopic behaviors of large neuronal networks from relevant descriptions of the dynamics of noisy neuronal networks remains today one of the main challenges in computational neuroscience, as discussed in P. Bressloff's review

As opposed to a large body of literature

The paper is organized as follows. In the Material and Methods section, we will introduce the basic network equations and their mean-field limits, and describe the methodology we propose for deriving macroscopic equations. We first show a rigorous derivation for the deterministic McKean neurons and numerically extend this to more general cases. This method reduces the dynamics of the average firing-rate to the knowledge of a particular function, the effective non-linearity, which can be numerically computed in all cases. This methodology is put in good use in the Results section in the case of the McKean, Fitzhugh-Nagumo and Hodgkin-Huxley neurons. In each case, the effective nonlinearity is numerically computed for different noise levels. The reduced low-dimensional macroscopic system is then compared to simulations of large networks, and will show a precise agreement. We also numerically investigate the robustness of the reduction with respect to the variation of parameters. The discussion section explores some implications of the present approach.

In this section we introduce the networks models considered, their mean-field limits and the formal derivation of the dynamics of averaged firing-rate models. This approach will be used in the result section to derive macroscopic limits and demonstrate the validity of the reduction. The python programs corresponding to the simulations can be downloaded at the url

We consider a network of P populations being composed of

In this equation, the functions

Three main models used in computational neuroscience addressed in the present manuscript are the McKean model

is probably the most widely accepted neuron model from the electrophysiological viewpoint. The Hodgkin-Huxley model describes the evolution of the membrane potential in relationship with the dynamics of ionic currents flowing across the cellular membrane of the neuron. It was introduced in the 1950s in

was introduced in

is a piecewise continuous approximation of the Fitzhugh-Nagumo model that presents the mathematical advantage of allowing explicit calculations and analytic developments

with

The behavior of very large random stochastic networks can be adequately described in the

In our model, we assume that the synaptic weights are randomly drawn from a normal law. Specifically, we consider that the connection

As shown in

The independence property (called propagation of chaos in Boltzmann's kinetic theory) ensure that in the limit, each neuron produces an independent realization of the same probability distribution, and thus, samples this law. Therefore, any statistics of the neuronal activity in a population can be accessed. An important example is the empirical average of the activity, which converges towards the expectation of the solution to the mean-field equation. This property has the important consequence that the averaged activity of all neurons in a population can be accessible through the mean-field equations.

Let us eventually notice that the mean-field

Now that we introduced the network models and the limits we are interested in, we are in a position to define the observable macroscopic quantity that will describe the activity of the network.

The averaged firing-rate is usually considered as a relevant macroscopic description of the population activity. Heuristically, this quantity corresponds to the number of spikes fired in a certain time window averaged over all neurons in the same population. Of course, counting discrete events is a non-trivial operation, and several computational definitions have been proposed

This complexity motivates the introduction of an analogous variable to the firing rate, which we call macroscopic activity, simply defined as the averaged membrane potential of neurons belonging to a given population and within a certain time window. Although this measure does not explicit count spikes, it is closely related to the firing-rate. Indeed, given that neurons communicate via spikes which are stereotyped electrical impulses of extreme amplitude, averaging the value of the membrane potential during a time window and dividing by the area under a spike provide a rough estimate of the number of spikes emitted. The main difference between the two measures is that it is affected by the subthreshold neuronal activity

In detail, we define the

This equation is not closed because of the term

In order to explain the principle of the reduction, we start by treating analytically the simplest case considered, namely the McKean neurons networks with no noise. In this model, we compute

We consider a McKean network (4) in the mean field limit (5) and further consider

In the sequel, we use an implicit integration of the adaptation variable

As said before, the only unknown term in the formula above is

Corresponds to

Under the assumption that the recovery variable is very slow (

with

We write

If

Then the system has a single stable fixed point on the negative (resp. positive) slow manifold. In

If

Then the system is oscillating on a deterministic limit cycle represented in orange in

Similarly,

Therefore, for

This function is shown in

Based on

In the general case (stochastic nonlinear neurons), one can numerically compute the effective non-linearity. To make sense of the term

the ansatz reads:

To evaluate

One of the pitfalls of this methods occurs if the mean-field equation present multiple stable stationary or periodic attractors. In that particular case, the quantity

Value of

Identifying

Model | Linear part |

McKean | |

Fitzhugh-Nagumo | |

Hodgkin-Huxley |

For McKean and Fitzhugh-Nagumo models, the convolution accounts entirely for the existence of the adaptation variable

For regime I neurons, when the effective non-linearity

For regime II neurons, when the initial condition is not in the bistable region, we will consider that the averaged system pursues on the initial attractor (fixed point or spiking cycle) when possible, and switches attractors if the activity brings the system in regions where the initial attractor disappears. In details, let us denote by

This approximation will be efficient if the probability to switch from one attractor to the other is small, e.g. for small noise.

In this section, we evaluate the accuracy of the reduced model presented above for the three neuron models McKean, Fitzhugh-Nagumo and Hodgkin-Huxley. First, we address the computation of the effective non-linearity both in the deterministic and noisy cases. Second, we confront the time course of the macroscopic activity calculated according to our reduction against the a posteriori average of the activity of a spiking network.

In the deterministic reduction (small noise limit), the effective non-linearity can be obtained through the bifurcation analysis of a single neuron. Indeed, the effective sigmoid amounts to computing the temporal average

The upper row shows the temporal average of the solutions (i.e. the fixed points and average value in the case of periodic orbits) and the lower row shows the frequency of the regular spiking regime, in the McKean model (left), Fitzhugh-Nagumo model (center) and Hodgkin-Huxley model (right).

Although the deterministic McKean neuron has been analytically treated previously, we now consider it under the angle of bifurcations for consistency with the other models. In the McKean neuron, the non-differentiable, piecewise-continuous nature of the flow gives rise to a non-smooth Hopf bifurcation associated with a branch of stable limit cycles. The emergence of the cycle arises through a non-smooth homoclinic bifurcation, hence corresponding to the existence of arbitrarily slow periodic orbits, typical of a class I excitability in the Hodgkin classification. In this model, an important distinction is the absence of bistable regime: the average variable has a unique value whatever the initial condition and whatever the input chosen. In the present case, stable permanent regimes are unique. Therefore, there exists a single-valued function

The bifurcation diagram of the Fitzhugh-Nagumo neuron as a function of the input level (see

This model displays qualitatively the same features as that of the Fitzhugh-Nagumo model but with significant quantitative differences. In particular, the bifurcation diagram of the Hodgkin-Huxley neuron (

In order to compute effective nonlinearities in the presence of noise, we resort to numerical simulations. The method described in Material and Methods provide the surfaces plotted in

Observe that noise tends to have a smoothing effect on the sigmoids.For the Hodgkin-Huxley model, we have empirically chosen a noise threshold under which the neuron was considered regime II and above which it is regime I. There are thus 2 branches below the threshold and only one above.

In the case of the Fitzhugh-Nagumo model, we observe that the regime II is not observed in simulations in the presence of noise. This is due to the smallness of the parameter region corresponding to the bistable regime, and the averaged system can be well approximated by regime I dynamics. In the case of the Hodgkin-Huxley network, there are clearly two different behaviors depending of the level of intrinsic noise. When the noise is small (resp. large) the neuron is regime II (resp. I). Interestingly, this shows how a strong noise can qualitatively simplify the macroscopic dynamics of a network.

We now simulate large networks of McKean, Fitzhugh-Nagumo and Hodgkin-Huxley neurons, compute numerically their macroscopic averaged activity, and compare the dynamics of this variable to simulations of the reduced ordinary differential equation involving our effective non-linearity function in all three cases.

We consider network models with

Averaged macroscopic variables are in plain lines and simulations of the macroscopic equations are in dashed lines. The variable related to the

The comparison for the McKean and Fitzhugh-Nagumo neurons show a precise match see

The comparison for the Hodgkin-Huxley neuron are only a relative success. The first difficulty arises, as expected, for small noise within regime II where the reduction is not univocally determined. The algorithm proposed in Material and Methods to simulate the regime II networks fails reproducing faithfully the averaged spiking network, see

In

The y-axis corresponds to the mean of the absolute value of the difference between the a posteriori averaged full system and the reduced system. (left)

The left picture in

However, the right picture shows that the slowness of the synapses is a critical feature enabling the reduction. The synapses have to be at least one order of magnitude slower than the neurons' activity. Yet, the fastest synapses enabling the reduction, i.e.

Even if collective phenomena arising in large noisy spiking neural networks are extremely complex, we have shown that, under some assumptions and for some models, a macroscopic variable describing the global behavior of the network can be consistently described by simple low dimensional deterministic differential equations. The parameters and non-linearity involved are determined by the type of neurons considered and by the level of noise neurons are subjected to. Depending on the neuron model the non-linearity can be a simple, well-behaved function (which we call regime I) or a more complicated multivalued function (which we call regime II in case of two values). The three neuron models we considered (McKean, Fitzhugh-Nagumo and Hodgkin-Huxley) are regime I when the intrinsic noise is strong in the network, and we expect this property to be valid for any type of neuron models. However, with weak noise the Hodgkin-Huxley model is regime II, in which case the low-dimensional model proposed is more complicated (involving jumps between attractors). Comparisons of the averaged dynamics of spiking networks with the reduced equations showed a very precise fit, even for initial conditions independent of the network initial conditions, for regime I neuron models. However, for the regime II neuron models, the reduction accuracy is not as good. Indeed, noise will induce random switches from one attractor to the other, which cannot be handled through reduced methods, and therefore path-wise fit are bound to be out of reach. Yet, the reduced model recovers the main qualitative features of the signal, but in the bistable regions, quantitative distinctions arise.

For neurons in regime I, the reduction accuracy is significantly better for McKean and Fitzhugh-Nagumo neuron models than for the Hodgkin-Huxley model. Indeed, the reduction for the latter becomes irrelevant for strong connections between neurons whereas it is not the case for the former. We believe this is not due an inherent difference between the models, but rather to an inadequacy of the choice of the linear part

This reduction relies on a number of assumptions imposed by the mathematical approach: first, the approach is valid when neuronal populations are large and randomly connected for averaging effects to occur (i.e. for the mean-field reduction to hold). More importantly, the reduction is largely based on the linearity of synapses. As said in the introduction, this assumption is not fully consistent with the biological system. It was however necessary to perform the reduction. Extending this approach to non-linear synapses is an important improvement to increase biological plausibility of these results well worth investigating. Another assumption was the slowness of synapses and inputs. This assumption, required in our mathematical developments, does not seem critical. Indeed, simulations have shown that the reduction was quite robust to increased speed for synapse and inputs.

It is important to note the extreme complexity reduction obtained: in the case of Fitzhugh-Nagumo networks, we reduced a system of

This study quantifies the stabilization properties of the noise, that were already discussed in

As opposed to former reduction techniques