This section sets the stage for our results. We review in the ‘Hodgkin-Huxley model’ section the Hodgkin-Huxley model equations in the case where both the membrane potential and the ion channel equations include noise. We then proceed in the ‘The FitzHugh-Nagumo model’ section with the FitzHugh-Nagumo equations in the case where the membrane potential equation includes noise. We next discuss in the ‘Models of synapses and maximum conductances’ section the connectivity models of networks of such neurons, starting with the synapses, electrical and chemical, and finishing with several stochastic models of the synaptic weights. In the ‘Putting everything together’ section, we write the network equations in the various cases considered in the previous section and express them in a general abstract mathematical form that is the one used for stating and proving the results about the mean-field limits in the ‘Mean-field equations for conductance-based models’ section. Before we jump into this, we conclude in the ‘Mean-field methods in computational neuroscience: a quick overview’ section with a brief overview of the mean-field methods popular in computational neuroscience.

From the mathematical point of view, each neuron is a complex system, whose dynamics is often described by a set of stochastic nonlinear differential equations. Such models aim at reproducing the biophysics of ion channels governing the membrane potential and therefore the spike emission. This is the case of the classical model of Hodgkin and Huxley [14] and of its reductions [15–17]. Simpler models use discontinuous processes mimicking the spike emission by modeling the membrane voltage and considering that spikes are emitted when it reaches a given threshold. These are called integrate-and-fire models [18, 19] and will not be addressed here. The models of large networks we deal with here therefore consist of systems of coupled nonlinear diffusion processes.

### 2.1 Hodgkin-Huxley model

One of the most important models in computational neuroscience is the Hodgkin-Huxley model. Using pioneering experimental techniques of that time, Hodgkin and Huxley [14] determined that the activity of the giant squid axon is controlled by three major currents: voltage-gated persistent {\mathrm{K}}^{+} current with four activation gates, voltage-gated transient {\mathrm{Na}}^{+} current with three activation gates and one inactivation gate, and Ohmic leak current, {I}_{\mathrm{L}}, which is carried mostly by chloride ions ({\mathrm{Cl}}^{-}). In this paper, we only use the space-clamped Hodgkin-Huxley model which we slightly generalize to a stochastic setting in order to better take into account the variability of the parameters. The advantages of this model are numerous, and one of the most prominent aspects in its favor is its correspondence with the most widely accepted formalism to describe the dynamics of the nerve cell membrane. A very extensive literature can also be found about the mathematical properties of this system, and it is now quite well understood.

The basic electrical relation between the membrane potential and the currents is simply:

C\frac{dV}{dt}={I}^{\mathrm{ext}}(t)-{I}_{\mathrm{K}}-{I}_{\mathrm{Na}}-{I}_{\mathrm{L}},

where {I}^{\mathrm{ext}}(t) is an external current. The detailed expressions for {I}_{\mathrm{K}}, {I}_{\mathrm{Na}} and {I}_{\mathrm{L}} can be found in several textbooks, e.g. [17, 20]:

\begin{array}{rcl}{I}_{\mathrm{K}}& =& {\overline{g}}_{\mathrm{K}}{n}^{4}(V-{E}_{\mathrm{K}}),\\ {I}_{\mathrm{Na}}& =& {\overline{g}}_{\mathrm{Na}}{m}^{3}h(V-{E}_{\mathrm{Na}}),\\ {I}_{\mathrm{L}}& =& {g}_{\mathrm{L}}(V-{E}_{\mathrm{L}}),\end{array}

where {\overline{g}}_{\mathrm{K}} (respectively, {\overline{g}}_{\mathrm{Na}}) is the maximum conductance of the potassium (respectively, the sodium) channel; {g}_{\mathrm{L}} is the conductance of the Ohmic channel; and *n* (respectively, *m*) is the activation variable for {\mathrm{K}}^{+} (respectively, for Na). There are four (respectively, three) activation gates for the {\mathrm{K}}^{+} (respectively, the Na) current which accounts for the power 4 (respectively, 3) in the expression of {I}_{\mathrm{K}} (respectively {I}_{\mathrm{Na}}). *h* is the inactivation variable for Na. These activation/deactivation variables, denoted by x\in \{n,m,h\} in what follows, represent a proportion (they vary between 0 and 1) of open gates. The proportions of open channels are given by the functions {n}^{4} and {m}^{3}h. The proportions of open gates can be computed through a Markov chain modeling assuming the gates to open with rate {\rho}_{x}(V) (the dependence in *V* accounts for the voltage-gating of the gate) and to close with rate {\zeta}_{x}(V). These processes can be shown to converge, under standard assumptions, towards the following ordinary differential equations:

\dot{x}={\rho}_{x}(V)(1-x)-{\zeta}_{x}(V)x,\phantom{\rule{1em}{0ex}}x\in \{n,m,h\}.

The functions {\rho}_{x}(V) and {\zeta}_{x}(V) are smooth functions whose exact values can be found in several textbooks such as the ones cited above. Note that half of these six functions are unbounded when the voltage goes to −∞, being of the form {k}_{1}{e}^{-{k}_{2}V}, with {k}_{1} and {k}_{2} as two positive constants. Since these functions have been fitted to experimental data corresponding to values of the membrane potential between roughly −100 and 100 mVs, it is clear that extremely large in magnitude and negative values of this variable do not have any physiological meaning. We can therefore safely, smoothly perturb these functions so that they are upper-bounded by some large (but finite) positive number for these values of the membrane potential. Hence, the functions {\rho}_{x} and {\zeta}_{x} are bounded and Lipschitz continuous for x\in \{n,m,h\}. A more precise model taking into account the finite number of channels through the Langevin approximation results in the stochastic differential equation^{Footnote 1}

d{x}_{t}=({\rho}_{x}(V)(1-x)-{\zeta}_{x}(V)x)\phantom{\rule{0.2em}{0ex}}dt+\sqrt{{\rho}_{x}(V)(1-x)+{\zeta}_{x}(V)x}\chi (x)\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{x},

where {W}_{t}^{x} and x\in \{n,m,h\} are independent standard Brownian motions. \chi (x) is a function that vanishes outside (0,1). This guarantees that the solution remains a proportion, i.e. lies between 0 and 1 for all times. We define

{\sigma}_{x}(V,x)=\sqrt{{\rho}_{x}(V)(1-x)+{\zeta}_{x}(V)x}\chi (x).

(1)

In order to complete our stochastic Hodgkin-Huxley neuron model, we assume that the external current {I}^{\mathrm{ext}}(t) is the sum of a deterministic part, noted as I(t), and a stochastic part, a white noise with variance {\sigma}_{\mathrm{ext}} built from a standard Brownian motion {W}_{t} independent of {W}_{t}^{x} and x\in \{n,m,h\}. Considering the current produced by the income of ion through these channels, we end up with the following system of stochastic differential equations:

\{\begin{array}{c}C\phantom{\rule{0.2em}{0ex}}d{V}_{t}=(I(t)-{\overline{g}}_{\mathrm{K}}{n}^{4}(V-{E}_{\mathrm{K}})-{\overline{g}}_{\mathrm{Na}}{m}^{3}h(V-{E}_{\mathrm{Na}})-{\overline{g}}_{\mathrm{L}}(V-{E}_{\mathrm{L}}))\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{Cd{V}_{t}=}+{\sigma}_{\mathrm{ext}}\phantom{\rule{0.2em}{0ex}}d{W}_{t},\hfill \\ d{x}_{t}=({\rho}_{x}(V)(1-x)-{\zeta}_{x}(V)x)\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{x}(V,x)\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{x},\phantom{\rule{1em}{0ex}}x\in \{n,m,h\}.\hfill \end{array}

(2)

This is a stochastic version of the Hodgkin-Huxley model. The functions {\rho}_{x} and {\zeta}_{x} are bounded and Lipschitz continuous (see discussion above). The functions *n*, *m* and *h* are bounded between 0 and 1; hence, the functions {n}^{4} and {m}^{3}h are Lipschitz continuous.

To illustrate the model, we show in Figure 1 the time evolution of the three ion channel variables *n*, *m* and *h* as well as that of the membrane potential *V* for a constant input I=20.0. The system of ordinary differential equations (ODEs) has been solved using a Runge-Kutta scheme of order 4 with an integration time step \mathrm{\Delta}t=0.01. In Figure 2, we show the same time evolution when noise is added to the channel variables and the membrane potential.

For the membrane potential, we have used {\sigma}_{\mathrm{ext}}=3.0 (see Equation 2), while for the noise in the ion channels, we have used the following *χ* function (see Equation 1):

\chi (x)=\{\begin{array}{cc}\mathrm{\Gamma}{e}^{-\mathrm{\Lambda}/(1-{(2x-1)}^{2})}\hfill & \text{if}0x1\hfill \\ 0\hfill & \text{if}x\le 0\vee x\ge 1\hfill \end{array}

(3)

with \mathrm{\Gamma}=0.1 and \mathrm{\Lambda}=0.5 for all the ion channels. The system of SDEs has been integrated using the Euler-Maruyama scheme with \mathrm{\Delta}t=0.01.

Because the Hodgkin-Huxley model is rather complicated and high-dimensional, many reductions have been proposed, in particular to two dimensions instead of four. These reduced models include the famous FitzHugh-Nagumo and Morris-Lecar models. These two models are two-dimensional approximations of the original Hodgkin-Huxley model based on quantitative observations of the time scale of the dynamics of each variable and identification of variables. Most reduced models still comply with the Lipschitz and linear growth conditions ensuring the existence and uniqueness of a solution, except for the FitzHugh-Nagumo model which we now introduce.

### 2.2 The FitzHugh-Nagumo model

In order to reduce the dimension of the Hodgkin-Huxley model, FitzHugh [15, 16, 21] introduced a simplified two-dimensional model. The motivation was to isolate conceptually essential mathematical features yielding excitation and transmission properties from the analysis of the biophysics of sodium and potassium flows. Nagumo and collaborators [22] followed up with an electrical system reproducing the dynamics of this model and studied its properties. The model consists of two equations, one governing a voltage-like variable *V* having a cubic nonlinearity and a slower recovery variable *w*. It can be written as:

\{\begin{array}{c}\dot{V}=f(V)-w+{I}^{\mathrm{ext}},\hfill \\ \dot{w}=c(V+a-bw),\hfill \end{array}

(4)

where f(V) is a cubic polynomial in *V* which we choose, without loss of generality, to be t(V)=V-{V}^{3}/3. The parameter {I}^{\mathrm{ext}} models the input current the neuron receives; the parameters *a*, b>0 and c>0 describe the kinetics of the recovery variable *w*. As in the case of the Hodgkin-Huxley model, the current {I}^{\mathrm{ext}} is assumed to be the sum of a deterministic part, noted *I*, and a stochastic white noise accounting for the randomness of the environment. The stochastic FitzHugh-Nagumo equation is deduced from Equation 4 and reads:

\{\begin{array}{c}d{V}_{t}=({V}_{t}-\frac{{V}_{t}^{3}}{3}-{w}_{t}+I)\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{\mathrm{ext}}\phantom{\rule{0.2em}{0ex}}d{W}_{t},\hfill \\ d{w}_{t}=c({V}_{t}+a-b{w}_{t})\phantom{\rule{0.2em}{0ex}}dt.\hfill \end{array}

(5)

Note that because the function f(V) is not *g* lobally Lipschitz continuous (only locally), the well-posedness of the stochastic differential equation (Equation 5) does not follow immediately from the standard theorem which assumes the global Lipschitz continuity of the drift and diffusion coefficients. This question is settled below by Proposition 1.

We show in Figure 3 the time evolution of the adaptation variable and the membrane potential in the case where the input *I* is constant and equal to 0.7. The left-hand side of the figure shows the case with no noise while the right-hand side shows the case where noise of intensity {\sigma}_{\mathrm{ext}}=0.25 (see Equation 5) has been added.

The deterministic model has been solved with a Runge-Kutta method of order 4, while the stochastic model, with the Euler-Maruyama scheme. In both cases, we have used an integration time step \mathrm{\Delta}t=0.01.

### 2.3 Partial conclusion

We have reviewed two main models of space-clamped single neurons: the Hodgkin-Huxley and FitzHugh-Nagumo models. These models are stochastic, including various sources of noise: external and internal. The noise sources are supposed to be independent Brownian processes. We have shown that the resulting stochastic differential Equations 2 and 5 were well-posed. As pointed out above, this analysis extends to a large number of reduced versions of the Hodgkin-Huxley such as those that can be found in the book [17].

### 2.4 Models of synapses and maximum conductances

We now study the situation in which several of these neurons are connected to one another forming a network, which we will assume to be fully connected. Let *N* be the total number of neurons. These neurons belong to *P* populations, e.g. pyramidal cells or interneurons. If the index of a neuron is *i*, 1\le i\le N, we note p(i)=\alpha, 1\le \alpha \le P as the population it belongs to. We note {N}_{p(i)} as the number of neurons in population p(i). Since we want to be as close to biology as possible while keeping the possibility of a mathematical analysis of the resulting model, we consider two types of simplified, but realistic, synapses: chemical and electrical or gap junctions. The following material concerning synapses is standard and can be found in textbooks [20]. The new, and we think important, twist is to add noise to our models. To unify notations, in what follows, *i* is the index of a postsynaptic neuron belonging to population \alpha =p(i), and *j* is the index of a presynaptic neuron to neuron *i* belonging to population \gamma =p(j).

#### 2.4.1 Chemical synapses

The principle of functioning of chemical synapses is based on the release of a neurotransmitter in the presynaptic neuron synaptic button, which binds to specific receptors on the postsynaptic cell. This process, similar to the currents described in the Hodgkin and Huxley model, is governed by the value of the cell membrane potential. We use the model described in [20, 23], which features a quite realistic biophysical representation of the processes at work in the spike transmission and is consistent with the previous formalism used to describe the conductances of other ion channels. The model emulates the fact that following the arrival of an action potential at the presynaptic terminal, a neurotransmitter is released in the synaptic cleft and binds to the postsynaptic receptor with a first order kinetic scheme. Let *j* be a presynaptic neuron to the postynaptic neuron *i*. The synaptic current induced by the synapse from *j* to *i* can be modelled as the product of a conductance {g}_{ij} with a voltage difference:

{I}_{ij}^{\mathrm{syn}}=-{g}_{ij}(t)({V}^{i}-{V}_{\mathrm{rev}}^{ij}).

(6)

The synaptic reversal potentials {V}_{\mathrm{rev}}^{ij} are approximately constant within each population: {V}_{\mathrm{rev}}^{ij}:={V}_{\mathrm{rev}}^{\alpha \gamma}. The conductance {g}_{ij} is the product of the maximum conductance {J}_{ij}(t) with a function {y}^{j}(t) that denotes the fraction of open channels and depends only upon the presynaptic neuron *j*:

{g}_{ij}(t)={J}_{ij}(t){y}^{j}(t).

(7)

The function {y}^{j}(t) is often modelled [20] as satisfying the following ordinary differential equation:

{\dot{y}}^{j}(t)={a}_{r}^{j}{S}_{j}\left({V}^{j}\right)(1-{y}^{j}(t))-{a}_{d}^{j}{y}^{j}(t).

The positive constants {a}_{r}^{j} and {a}_{d}^{j} characterize the rise and decay rates, respectively, of the synaptic conductance. Their values depend only on the population of the presynaptic neuron *j*, i.e. {a}_{r}^{j}:={a}_{r}^{\gamma} and {a}_{d}^{j}:={a}_{d}^{\gamma}, but may vary significantly from one population to the next. For example, gamma-aminobutyric acid {\text{(GABA)}}_{\mathrm{B}} synapses are slow to activate and slow to turn off while the reverse is true for {\mathrm{GABA}}_{\mathrm{A}} and AMPA synapses [20]. {S}_{j}({V}^{j}) denotes the concentration of the transmitter released into the synaptic cleft by a presynaptic spike. We assume that the function {S}_{j} is sigmoidal and that its exact form depends only upon the population of the neuron *j*. Its expression is given by (see, e.g. [20]):

{S}_{\gamma}\left({V}^{j}\right)=\frac{{T}_{\mathrm{max}}^{\gamma}}{1+{e}^{-{\lambda}_{\gamma}({V}^{j}-{V}_{T}^{\gamma})}}.

(8)

Destexhe et al. [23] give some typical values of the parameters {T}_{\mathrm{max}}=1\text{mM}, {V}_{T}=2\text{mV} and 1/\lambda =5\text{mV}.

Because of the dynamics of ion channels and of their finite number, similar to the channel noise models derived through the Langevin approximation in the Hodgkin-Huxley model (Equation 2), we assume that the proportion of active channels is actually governed by a stochastic differential equation with diffusion coefficient {\sigma}_{\gamma}(V,y) depending only on the population *γ* of *j* of the form (Equation 1):

d{y}_{t}^{j}=({a}_{r}^{\gamma}{S}_{\gamma}\left({V}^{j}\right)(1-{y}^{j}(t))-{a}_{d}^{\gamma}{y}^{j}(t))\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{\gamma}^{y}({V}^{j},{y}^{j})\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{j,y}.

In detail, we have

{\sigma}_{\gamma}^{y}({V}^{j},{y}^{j})=\sqrt{{a}_{r}^{\gamma}{S}_{\gamma}\left({V}^{j}\right)(1-{y}^{j})+{a}_{d}^{\gamma}{y}^{j}}\chi \left({y}^{j}\right).

(9)

Remember that the form of the diffusion term guarantees that the solutions to this equation with appropriate initial conditions stay between 0 and 1. The Brownian motions {W}^{j,y} are assumed to be independent from one neuron to the next.

#### 2.4.2 Electrical synapses

The electrical synapse transmission is rapid and stereotyped and is mainly used to send simple depolarizing signals for systems requiring the fastest possible response. At the location of an electrical synapse, the separation between two neurons is very small (≈3.5 nm). This narrow gap is bridged by the *gap junction channels*, specialized protein structures that conduct the flow of ionic current from the presynaptic to the postsynaptic cell (see, e.g. [24]).

Electrical synapses thus work by allowing ionic current to flow passively through the gap junction pores from one neuron to another. The usual source of this current is the potential difference generated locally by the action potential. Without the need for receptors to recognize chemical messengers, signaling at electrical synapses is more rapid than that which occurs across chemical synapses, the predominant kind of junctions between neurons. The relative speed of electrical synapses also allows for many neurons to fire synchronously.

We model the current for this type of synapse as

{I}_{ij}^{\mathrm{che}}={J}_{ij}(t)({V}^{i}-{V}^{j}),

(10)

where {J}_{ij}(t) is the maximum conductance.

#### 2.4.3 The maximum conductances

As shown in Equations 6, 7 and 10, we model the current going through the synapse connecting neuron *j* to neuron *i* as being proportional to the maximum conductance {J}_{ij}. Because the synaptic transmission through a synapse is affected by the nature of the environment, the maximum conductances are affected by dynamical random variations (we do not take into account such phenomena as plasticity). What kind of models can we consider for these random variations?

The simplest idea is to assume that the maximum conductances are independent diffusion processes with mean \frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}} and standard deviation \frac{{\sigma}_{\alpha \gamma}^{J}}{{N}_{\gamma}}, i.e. that depend only on the populations. The quantities {\overline{J}}_{\alpha \gamma}, being conductances, are positive. We write the following equation:

{J}_{i\gamma}(t)=\frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}+\frac{{\sigma}_{\alpha \gamma}^{J}}{{N}_{\gamma}}{\xi}^{i,\gamma}(t),

(11)

where the {\xi}^{i,\gamma}(t), i=1,\dots ,N, \gamma =1,\dots ,P, are *NP*-independent zero mean unit variance white noise processes derived from *NP*-independent standard Brownian motions {B}^{i,\gamma}(t), i.e. {\xi}^{i,\gamma}(t)=\frac{d{B}^{i,\gamma}(t)}{dt}, which we also assume to be independent of all the previously defined Brownian motions. The main advantage of this dynamics is its simplicity. Its main disadvantage is that if we increase the noise level {\sigma}_{\alpha \gamma}, the probability that {J}_{ij}(t) becomes negative increases also: this would result in a negative conductance!

One way to alleviate this problem is to modify the dynamics (Equation 11) to a slightly more complicated one whose solutions do not change sign, such as for instance, the Cox-Ingersoll-Ross model [25] given by:

d{J}_{ij}(t)={\theta}_{\alpha \gamma}(\frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}-{J}_{ij}(t))\phantom{\rule{0.2em}{0ex}}dt+\frac{{\sigma}_{\alpha \gamma}^{J}}{{N}_{\gamma}}\sqrt{{J}_{ij}(t)}\phantom{\rule{0.2em}{0ex}}d{B}^{i,\gamma}(t).

(12)

Note that the right-hand side only depends upon the population \gamma =p(j). Let {J}_{ij}(0) be the initial condition, it is known [25] that

\begin{array}{rcl}\mathbb{E}[{J}_{ij}(t)]& =& {J}_{ij}(0){e}^{-{\theta}_{\alpha \gamma}t}+\frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}(1-{e}^{-{\theta}_{\alpha \gamma}t}),\\ Var({J}_{ij}(t))& =& {J}_{ij}(0)\frac{{({\sigma}_{\alpha \gamma}^{J})}^{2}}{{N}_{\gamma}^{2}{\theta}_{\alpha \gamma}}({e}^{-{\theta}_{\alpha \gamma}t}-{e}^{-2{\theta}_{\alpha \gamma}t})+\frac{{\overline{J}}_{\alpha \gamma}{({\sigma}_{\alpha \gamma}^{J})}^{2}}{2{N}_{\gamma}^{3}{\theta}_{\alpha \gamma}}{(1-{e}^{-{\theta}_{\alpha \gamma}t})}^{2}.\end{array}

This shows that if the initial condition {J}_{ij}(0) is equal to the mean \frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}, the mean of the process is constant over time and equal to \frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}. Otherwise, if the initial condition {J}_{ij}(0) is of the same sign as {\overline{J}}_{\alpha \gamma}, i.e. positive, then the long term mean is \frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}} and the process is guaranteed not to touch 0 if the condition 2{N}_{\gamma}{\theta}_{\alpha \gamma}{\overline{J}}_{\alpha \gamma}\ge {({\sigma}_{\alpha \gamma}^{J})}^{2} holds [25]. Note that the long term variance is \frac{{\overline{J}}_{\alpha \gamma}{({\sigma}_{\alpha \gamma}^{J})}^{2}}{2{N}_{\gamma}^{3}{\theta}_{\alpha \gamma}}.

### 2.5 Putting everything together

We are ready to write the equations of a network of Hodgkin-Huxley or FitzHugh-Nagumo neurons and study their properties and their limit, if any, when the number of neurons becomes large. The external current for neuron *i* has been modelled as the sum of a deterministic part and a stochastic part:

{I}_{i}^{\mathrm{ext}}(t)={I}_{i}(t)+{\sigma}_{\mathrm{ext}}^{i}\frac{d{W}_{t}^{i}}{dt}.

We will assume that the deterministic part is the same for all neurons in the same population, {I}_{i}:={I}_{\alpha}, and that the same is true for the variance, {\sigma}_{\mathrm{ext}}^{i}:={\sigma}_{\mathrm{ext}}^{\alpha}. We further assume that the *N* Brownian motions {W}_{t}^{i} are *N*-independent Brownian motions and independent of all the other Brownian motions defined in the model. In other words,

{I}_{i}^{\mathrm{ext}}(t)={I}_{\alpha}(t)+{\sigma}_{\mathrm{ext}}^{\alpha}\frac{d{W}_{t}^{i}}{dt},\phantom{\rule{1em}{0ex}}\alpha =p(i),i=1,\dots ,N.

(13)

We only cover the case of chemical synapses and leave it to the reader to derive the equations in the simpler case of gap junctions.

#### 2.5.1 Network of FitzHugh-Nagumo neurons

We assume that the parameters {a}_{i}, {b}_{i} and {c}_{i} in Equation 5 of the adaptation variable {w}^{i} of neuron *i* are only functions of the population \alpha =p(i).

*Simple maximum conductance variation.* If we assume that the maximum conductances fluctuate according to Equation 11, the state of the *i* th neuron in a fully connected network of FitzHugh-Nagumo neurons with chemical synapses is determined by the variables ({V}^{i},{w}^{i},{y}^{i}) that satisfy the following set of 3*N* stochastic differential equations:

\{\begin{array}{c}d{V}_{t}^{i}=({V}_{t}^{i}-\frac{{({V}_{t}^{i})}^{3}}{3}-{w}_{t}^{i}+{I}^{\alpha}(t))\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{d{V}_{t}^{i}=}-\left(\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\sum _{j,p(j)=\gamma}{\overline{J}}_{\alpha \gamma}({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j}\right)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{d{V}_{t}^{i}=}-\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\left(\sum _{j,p(j)=\gamma}{\sigma}_{\alpha \gamma}^{J}({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j}\right)\phantom{\rule{0.2em}{0ex}}d{B}_{t}^{i,\gamma}\hfill \\ \phantom{d{V}_{t}^{i}=}+{\sigma}_{\mathrm{ext}}^{\alpha}\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i},\hfill \\ d{w}_{t}^{i}={c}_{\alpha}({V}_{t}^{i}+{a}_{\alpha}-{b}_{\alpha}{w}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt,\hfill \\ d{y}_{t}^{i}=({a}_{r}^{\alpha}{S}_{\alpha}\left({V}_{t}^{i}\right)(1-{y}_{t}^{i})-{a}_{d}^{\alpha}{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{\alpha}^{y}({V}_{t}^{i},{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i,y}.\hfill \end{array}

(14)

{S}_{\alpha}({V}_{t}^{i})

is given by Equation 8; {\sigma}_{\alpha}^{y}, by Equation 9; and {W}_{t}^{i,y}, i=1,\dots ,N, are *N*-independent Brownian processes that model noise in the process of transmitter release into the synaptic clefts.

*Sign-preserving maximum conductance variation.* If we assume that the maximum conductances fluctuate according to Equation 12, the situation is slightly more complicated. In effect, the state space of the neuron *i* has to be augmented by the *P* maximum conductances {J}_{i\gamma}, \gamma =1,\dots ,P. We obtain

\{\begin{array}{c}d{V}_{t}^{i}=({V}_{t}^{i}-\frac{{({V}_{t}^{i})}^{3}}{3}-{w}_{t}^{i}+{I}^{\alpha}(t))\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{d{V}_{t}^{i}=}-(\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\sum _{j,p(j)=\gamma}{J}_{ij}(t)({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j})\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{d{V}_{t}^{i}=}+{\sigma}_{\mathrm{ext}}^{\alpha}\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i},\hfill \\ d{w}_{t}^{i}={c}_{\alpha}({V}_{t}^{i}+{a}_{\alpha}-{b}_{\alpha}{w}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt,\hfill \\ d{y}_{t}^{i}=({a}_{r}^{\alpha}{S}_{\alpha}\left({V}_{t}^{i}\right)(1-{y}_{t}^{i})-{a}_{d}^{\alpha}{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{\alpha}^{y}({V}_{t}^{i},{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i,y},\hfill \\ d{J}_{i\gamma}(t)={\theta}_{\alpha \gamma}(\frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}-{J}_{i\gamma}(t))\phantom{\rule{0.2em}{0ex}}dt+\frac{{\sigma}_{\alpha \gamma}^{J}}{{N}_{\gamma}}\sqrt{{J}_{i\gamma}(t)}\phantom{\rule{0.2em}{0ex}}d{B}^{i,\gamma}(t),\phantom{\rule{1em}{0ex}}\gamma =1,\dots ,P,\hfill \end{array}

(15)

which is a set of N(P+3) stochastic differential equations.

#### 2.5.2 Network of Hodgkin-Huxley neurons

We provide a similar description in the case of the Hodgkin-Huxley neurons. We assume that the functions {\rho}_{x}^{i} and {\zeta}_{x}^{i}, x\in \{n,m,h\}, that appear in Equation 2 only depend upon \alpha =p(i).

*Simple maximum conductance variation.* If we assume that the maximum conductances fluctuate according to Equation 11, the state of the *i* th neuron in a fully connected network of Hodgkin-Huxley neurons with chemical synapses is therefore determined by the variables ({V}^{i},{n}^{i},{m}^{i},{h}^{i},{y}^{i}) that satisfy the following set of 5*N* stochastic differential equations:

\{\begin{array}{c}C\phantom{\rule{0.2em}{0ex}}d{V}_{t}^{i}=({I}^{\alpha}(t)-\overline{{g}_{\mathrm{K}}}{n}_{i}^{4}({V}_{t}^{i}-{E}_{\mathrm{K}})-\overline{{g}_{\mathrm{Na}}}{m}_{i}^{3}{h}_{i}({V}_{t}^{i}-{E}_{\mathrm{Na}})-\overline{{g}_{\mathrm{L}}}({V}_{t}^{i}-{E}_{\mathrm{L}}))\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{C\phantom{\rule{0.2em}{0ex}}d{V}_{t}^{i}=}-\left(\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\sum _{j,p(j)=\gamma}{\overline{J}}_{\alpha \gamma}({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j}\right)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{C\phantom{\rule{0.2em}{0ex}}d{V}_{t}^{i}=}-\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\left(\sum _{j,p(j)=\gamma}{\sigma}_{\alpha \gamma}^{J}({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j}\right)\phantom{\rule{0.2em}{0ex}}d{B}_{t}^{i,\gamma}\hfill \\ \phantom{C\phantom{\rule{0.2em}{0ex}}d{V}_{t}^{i}=}+{\sigma}_{\mathrm{ext}}^{\alpha}\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i},\hfill \\ d{x}_{i}(t)=({\rho}_{x}^{\alpha}\left({V}^{i}\right)(1-{x}_{i})-{\zeta}_{x}\left({V}^{i}\right){x}_{i})\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{x}({V}^{i},{x}_{i})\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{x,i},\phantom{\rule{1em}{0ex}}x\in \{n,m,h\},\hfill \\ d{y}_{t}^{i}=({a}_{r}^{\alpha}{S}_{\alpha}\left({V}_{t}^{i}\right)(1-{y}_{t}^{i})-{a}_{d}^{\alpha}{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{\alpha}^{y}({V}_{t}^{i},{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i,y}.\hfill \end{array}

(16)

*Sign-preserving maximum conductance variation.* If we assume that the maximum conductances fluctuate according to Equation 12, we use the same idea as in the FitzHugh-Nagumo case of augmenting the state space of each individual neuron and obtain the following set of (5+P)N stochastic differential equations:

\{\begin{array}{c}C\phantom{\rule{0.2em}{0ex}}d{V}_{t}^{i}=({I}^{\alpha}(t)-\overline{{g}_{\mathrm{K}}}{n}_{i}^{4}({V}_{t}^{i}-{E}_{\mathrm{K}})-\overline{{g}_{\mathrm{N}a}}{m}_{i}^{3}{h}_{i}({V}_{t}^{i}-{E}_{\mathrm{Na}})-\overline{{g}_{\mathrm{L}}}({V}_{t}^{i}-{E}_{\mathrm{L}}))\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{C\phantom{\rule{0.2em}{0ex}}d{V}_{t}^{i}=}-(\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\sum _{j,p(j)=\gamma}{J}_{ij}(t)({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j})\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{C\phantom{\rule{0.2em}{0ex}}d{V}_{t}^{i}=}+{\sigma}_{\mathrm{ext}}^{\alpha}\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i},\hfill \\ d{x}_{i}(t)=({\rho}_{x}^{\alpha}\left({V}_{t}^{i}\right)(1-{x}_{i})-{\zeta}_{x}^{\alpha}\left({V}_{t}^{i}\right){x}_{i})\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{x}({V}_{t}^{i},{x}_{i})\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{x,i},\phantom{\rule{1em}{0ex}}x\in \{n,m,h\},\hfill \\ d{y}_{t}^{i}=({a}_{r}^{\alpha}{S}_{\alpha}\left({V}_{t}^{i}\right)(1-{y}_{t}^{i})-{a}_{d}^{\alpha}{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{\sigma}_{\alpha}^{y}({V}_{t}^{i},{y}_{t}^{i})\phantom{\rule{0.2em}{0ex}}d{W}_{t}^{i,y},\hfill \\ d{J}_{i\gamma}(t)={\theta}_{\alpha \gamma}(\frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}-{J}_{i\gamma}(t))\phantom{\rule{0.2em}{0ex}}dt+\frac{{\sigma}_{\alpha \gamma}^{J}}{{N}_{\gamma}}\sqrt{{J}_{i\gamma}(t)}\phantom{\rule{0.2em}{0ex}}d{B}^{i,\gamma}(t),\phantom{\rule{1em}{0ex}}\gamma =1,\dots ,P.\hfill \end{array}

(17)

#### 2.5.3 Partial conclusion

Equations 14 to 17 have a quite similar structure. They are well-posed, i.e. given any initial condition, and any time T>0, they have a unique solution on [0,T] which is square-integrable. A little bit of care has to be taken when choosing these initial conditions for some of the parameters, i.e. *n*, *m* and *h*, which take values between 0 and 1, and the maximum conductances when one wants to preserve their signs.

In order to prepare the grounds for the ‘Mean-field equations for conductance-based models’ section, we explore a bit more the aforementioned common structure. Let us first consider the case of the simple maximum conductance variations for the FitzHugh-Nagumo network. Looking at Equation 14, we define the three-dimensional state vector of neuron *i* to be {X}_{t}^{i}=({V}_{t}^{i},{w}_{t}^{i},{y}_{t}^{i}). Let us now define {f}_{\alpha}:\mathbb{R}\times {\mathbb{R}}^{3}\to {\mathbb{R}}^{3}, \alpha =1,\dots ,P, by

{f}_{\alpha}(t,{X}_{t}^{i})=\left[\begin{array}{c}{V}_{t}^{i}-\frac{{({V}_{t}^{i})}^{3}}{3}-{w}_{t}^{i}+{I}^{\alpha}(t)\\ {c}_{\alpha}({V}_{t}^{i}+{a}_{\alpha}-{b}_{\alpha}{w}_{t}^{i})\\ {a}_{r}^{\alpha}{S}_{\alpha}\left({V}_{t}^{i}\right)(1-{y}_{t}^{i})-{a}_{d}^{\alpha}{y}_{t}^{i}\end{array}\right].

Let us next define {g}_{\alpha}:\mathbb{R}\times {\mathbb{R}}^{3}\to {\mathbb{R}}^{3\times 2} by

{g}_{\alpha}(t,{X}_{t}^{i})=\left[\begin{array}{cc}{\sigma}_{\mathrm{ext}}^{\alpha}& 0\\ 0& 0\\ 0& {\sigma}_{\alpha}^{y}({V}_{t}^{i},{y}_{t}^{i})\end{array}\right].

It appears that the intrinsic dynamics of the neuron *i* is conveniently described by the equation

d{X}_{t}^{i}={f}_{\alpha}(t,{X}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{g}_{\alpha}(t,{X}_{t}^{i})\left[\begin{array}{c}d{W}_{t}^{i}\\ d{W}_{t}^{i,y}\end{array}\right].

We next define the functions {b}_{\alpha \gamma}:{\mathbb{R}}^{3}\times {\mathbb{R}}^{3}\to {\mathbb{R}}^{3}, for \alpha ,\gamma =1,\dots ,P, by

{b}_{\alpha \gamma}({X}_{t}^{i},{X}_{t}^{j})=\left[\begin{array}{c}-{\overline{J}}_{\alpha \gamma}({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j}\\ 0\\ 0\end{array}\right]

and the function {\beta}_{\alpha \gamma}:{\mathbb{R}}^{3}\times {\mathbb{R}}^{3}\to {\mathbb{R}}^{3\times 1} by

{\beta}_{\alpha \gamma}({X}_{t}^{i},{X}_{t}^{j})=\left[\begin{array}{c}-{\sigma}_{\alpha \gamma}^{J}({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j}\\ 0\\ 0\end{array}\right].

It appears that the full dynamics of the neuron *i*, corresponding to Equation 14, can be described compactly by

\begin{array}{rcl}d{X}_{t}^{i}& =& {f}_{\alpha}(t,{X}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{g}_{\alpha}(t,{X}_{t}^{i})\left[\begin{array}{c}d{W}_{t}^{i}\\ d{W}_{t}^{i,y}\end{array}\right]+\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\sum _{j,p(j)=\gamma}{b}_{\alpha \gamma}({X}_{t}^{i},{X}_{t}^{j})\phantom{\rule{0.2em}{0ex}}dt\\ +\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\sum _{j,p(j)=\gamma}{\beta}_{\alpha \gamma}({X}_{t}^{i},{X}_{t}^{j})\phantom{\rule{0.2em}{0ex}}d{B}_{t}^{i,\gamma}.\end{array}

(18)

Let us now move to the case of the sign-preserving variation of the maximum conductances, still for the FitzHugh-Nagumo neurons. The state of each neuron is now *P*+3-dimensional: we define {X}_{t}^{i}=({V}_{t}^{i},{w}_{t}^{i},{y}_{t}^{i},{J}_{i1}(t),\dots ,{J}_{iP}(t)). We then define the functions {f}_{\alpha}:\mathbb{R}\times {\mathbb{R}}^{P+3}\to {\mathbb{R}}^{P+3}, \alpha =1,\dots ,P, by

{f}_{\alpha}(t,{X}_{t}^{i})=\left[\begin{array}{c}{V}_{t}^{i}-\frac{{({V}_{t}^{i})}^{3}}{3}-{w}_{t}^{i}+{I}^{\alpha}(t)\\ {c}_{\alpha}({V}_{t}^{i}+{a}_{\alpha}-{b}_{\alpha}{w}_{t}^{i})\\ {a}_{r}^{\alpha}{S}_{\alpha}\left({V}_{t}^{i}\right)(1-{y}_{t}^{i})-{a}_{d}^{\alpha}{y}_{t}^{i}\\ {\theta}_{\alpha \gamma}(\frac{{\overline{J}}_{\alpha \gamma}}{{N}_{\gamma}}-{J}_{i\gamma}(t)),\gamma =1,\dots ,P\end{array}\right]

and the functions {g}_{\alpha}:\mathbb{R}\times {\mathbb{R}}^{P+3}\to {\mathbb{R}}^{(P+3)\times (P+2)} by

{g}_{\alpha}(t,{X}_{t}^{i})=\left[\begin{array}{ccccc}{\sigma}_{\mathrm{ext}}^{\alpha}& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ 0& {\sigma}_{\alpha}^{y}({V}_{t}^{i},{y}_{t}^{i})& 0& \cdots & 0\\ 0& 0& \frac{{\sigma}_{\alpha 1}^{J}}{{N}_{1}}\sqrt{{J}_{i1}(t)}& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0& 0& 0& \cdots & \frac{{\sigma}_{\alpha P}^{J}}{{N}_{P}}\sqrt{{J}_{iP}(t)}\end{array}\right].

It appears that the intrinsic dynamics of the neuron *i* isolated from the other neurons is conveniently described by the equation

d{X}_{t}^{i}={f}_{\alpha}(t,{X}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{g}_{\alpha}(t,{X}_{t}^{i})\left[\begin{array}{c}d{W}_{t}^{i}\\ d{W}_{t}^{i,y}\\ d{B}_{t}^{i,1}\\ \vdots \\ d{B}_{t}^{i,P}\end{array}\right].

Let us finally define the functions {b}_{\alpha \gamma}:{\mathbb{R}}^{P+3}\times {\mathbb{R}}^{P+3}\to {\mathbb{R}}^{P+3}, for \alpha ,\gamma =1,\dots ,P, by

{b}_{\alpha \gamma}({X}_{t}^{i},{X}_{t}^{j})=\left[\begin{array}{c}-{J}_{ij}(t)({V}_{t}^{i}-{V}_{\mathrm{rev}}^{\alpha \gamma}){y}_{t}^{j}\\ 0\\ \vdots \\ 0\end{array}\right].

It appears that the full dynamics of the neuron *i*, corresponding to Equation 15 can be described compactly by

\begin{array}{rcl}d{X}_{t}^{i}& =& {f}_{\alpha}(t,{X}_{t}^{i})\phantom{\rule{0.2em}{0ex}}dt+{g}_{\alpha}(t,{X}_{t}^{i})\left[\begin{array}{c}d{W}_{t}^{i}\\ d{W}_{t}^{i,y}\\ d{B}_{t}^{i,1}\\ \vdots \\ d{B}_{t}^{i,P}\end{array}\right]\\ +\sum _{\gamma =1}^{P}\frac{1}{{N}_{\gamma}}\sum _{j,p(j)=\gamma}{b}_{\alpha \gamma}({X}_{t}^{i},{X}_{t}^{j})\phantom{\rule{0.2em}{0ex}}dt.\end{array}

(19)

We let the reader apply the same machinery to the network of Hodgkin-Huxley neurons.

Let us note *d* as the positive integer equal to the dimension of the state space in Equation 18 (d=3) or 19 (d=3+P) or in the corresponding cases for the Hodgkin-Huxley model (d=5 and d=5+P). The reader will easily check that the following four assumptions hold for both models:

(H1) *Locally Lipschitz dynamics*: For all \alpha \in \{1,\dots ,P\}, the functions {f}_{\alpha} and {g}_{\alpha} are uniformly locally Lipschitz continuous with respect to the second variable. In detail, for all U>0, there exists {K}_{U}>0 independent of t\in [0,T] such that for all x,y\in {B}_{U}^{d}, the ball of {\mathbb{R}}^{d} of radius *U*:

\parallel {f}_{\alpha}(t,x)-{f}_{\alpha}(t,y)\parallel +\parallel {g}_{\alpha}(t,x)-{g}_{\alpha}(t,y)\parallel \le {K}_{U}\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\alpha =1,\dots ,P.

(H2) *Locally Lipschitz interactions*: For all \alpha ,\gamma \in \{1,\dots ,P\}, the functions {b}_{\alpha \gamma} and {\beta}_{\alpha \gamma} are locally Lipschitz continuous. In detail, for all U>0, there exists {L}_{U}>0 such that for all x,y,{x}^{\prime},{y}^{\prime}\in {B}_{U}^{d}, we have:

\begin{array}{r}\parallel {b}_{\alpha \gamma}(x,y)-{b}_{\alpha \gamma}({x}^{\prime},{y}^{\prime})\parallel +\parallel {\beta}_{\alpha \gamma}(x,y)-{\beta}_{\alpha \gamma}({x}^{\prime},{y}^{\prime})\parallel \\ \phantom{\rule{1em}{0ex}}\le {L}_{U}(\parallel x-{x}^{\prime}\parallel +\parallel y-{y}^{\prime}\parallel ).\end{array}

(H3) *Linear growth of the interactions*: There exists a \tilde{K}>0 such that

max({\parallel {b}_{\alpha \gamma}(x,z)\parallel}^{2},{\parallel {\beta}_{\alpha \gamma}(x,z)\parallel}^{2})\le \tilde{K}(1+{\parallel x\parallel}^{2}).

(H4) *Monotone growth of the dynamics*: We assume that {f}_{\alpha} and {g}_{\alpha} satisfy the following monotonous condition for all \alpha =1,\dots ,P:

{x}^{T}{f}_{\alpha}(t,x)+\frac{1}{2}{\parallel {g}_{\alpha}(t,x)\parallel}^{2}\le K(1+{\parallel x\parallel}^{2}).

(20)

These assumptions are central to the proofs of Theorems 2 and 4.

They imply the following proposition stating that the system of stochastic differential equations (Equation 19) is well-posed:

**Proposition 1** *Let* T>0 *be a fixed time*. *If* |{I}^{\alpha}(t)|\le {I}_{m} *on* [0,T], *for* \alpha =1,\dots ,P, *Equations * 18 *and* 19 *together with an initial condition* {X}_{0}^{i}\in {\mathbb{L}}^{2}({\mathbb{R}}^{d}), i=1,\dots ,N *of square*-*integrable random variables*, *have a unique strong solution which belongs to* {\mathrm{L}}^{2}([0,T];{\mathbb{R}}^{dN}).

*Proof* The proof uses Theorem 3.5 in chapter 2 in [26] whose conditions are easily shown to follow from hypotheses 2.5.3 to (H2). □

The case N=1 implies that Equations 2 and 5, describing the stochastic FitzHugh-Nagumo and Hodgkin-Huxley neurons, are well-posed.

We are interested in the behavior of the solutions of these equations as the number of neurons tends to infinity. This problem has been long-standing in neuroscience, arousing the interest of many researchers in different domains. We discuss the different approaches developed in the field in the next subsection.

### 2.6 Mean-field methods in computational neuroscience: a quick overview

Obtaining the equations of evolution of the effective mean-field from microscopic dynamics is a very complex problem. Many approximate solutions have been provided, mostly based on the statistical physics literature.

Many models describing the emergent behavior arising from the interaction of neurons in large-scale networks have relied on continuum limits ever since the seminal work of Amari, and Wilson and Cowan [27–30]. Such models represent the activity of the network by macroscopic variables, e.g. the population-averaged firing rate, which are generally assumed to be deterministic. When the spatial dimension is not taken into account in the equations, they are referred to as neural masses, otherwise as neural fields. The equations that relate these variables are ordinary differential equations for neural masses and integrodifferential equations for neural fields. In the second case, they fall in a category studied in [31] or can be seen as ordinary differential equations defined on specific functional spaces [32]. Many analytical and numerical results have been derived from these equations and related to cortical phenomena, for instance, for the problem of spatio-temporal pattern formation in spatially extended models (see, e.g. [33–36]). The use of bifurcation theory has also proven to be quite powerful [37, 38]. Despite all its qualities, this approach implicitly makes the assumption that the effect of noise vanishes at the mesoscopic and macroscopic scales and hence that the behavior of such populations of neurons is deterministic.

A different approach has been to study regimes where the activity is uncorrelated. A number of computational studies on the integrate-and-fire neuron showed that under certain conditions, neurons in large assemblies end up firing asynchronously, producing null correlations [39–41]. In these regimes, the correlations in the firing activity decrease towards zero in the limit where the number of neurons tends to infinity. The emergent global activity of the population in this limit is deterministic and evolves according to a mean-field firing rate equation. However, according to the theory, these states only exist in the limit where the number of neurons is infinite, thereby raising the question of how the finiteness of the number of neurons impacts the existence and behavior of asynchronous states. The study of finite-size effects for asynchronous states is generally not reduced to the study of mean firing rates and can include higher order moments of firing activity [42–44]. In order to go beyond asynchronous states and take into account the stochastic nature of the firing and understand how this activity scales as the network size increases, different approaches have been developed, such as the population density method and related approaches [45]. Most of these approaches involve expansions in terms of the moments of the corresponding random variables, and the moment hierarchy needs to be truncated which is not a simple task that can raise a number of difficult technical issues (see, e.g. [46]).

However, increasingly many researchers now believe that the different intrinsic or extrinsic noise sources are part of the neuronal signal, and rather than being a pure disturbing effect related to the intrinsically noisy biological substrate of the neural system, they suggest that noise conveys information that can be an important principle of brain function [47]. At the level of a single cell, various studies have shown that the firing statistics are highly stochastic with probability distributions close to the Poisson distributions [48], and several computational studies confirmed the stochastic nature of single-cell firings [49–51]. How the variability at the single-neuron level affects the dynamics of cortical networks is less well established. Theoretically, the interaction of a large number of neurons that fire stochastic spike trains can naturally produce correlations in the firing activity of the population. For instance, power laws in the scaling of avalanche-size distributions has been studied both via models and experiments [52–55]. In these regimes, the randomness plays a central role.

In order to study the effect of the stochastic nature of the firing in large networks, many authors strived to introduce randomness in a tractable form. Some of the models proposed in the area are based on the definition of a Markov chain governing the firing dynamics of the neurons in the network, where the transition probability satisfies a differential equation, the *master equation*. Seminal works of the application of such modeling for neuroscience date back to the early 1990s and have been recently developed by several authors [43, 56–59]. Most of these approaches are proved correct in some parameter regions using statistical physics tools such as path integrals and Van-Kampen expansions, and their analysis often involve a moment expansion and truncation. Using a different approach, a static mean-field study of multi-population network activity was developed by Treves in [60]. This author did not consider external inputs but incorporated dynamical synaptic currents and adaptation effects. His analysis was completed in [39], where the authors proved, using a Fokker-Planck formalism, the stability of an asynchronous state in the network. Later on, Gerstner in [61] built a new approach to characterize the mean-field dynamics for the spike response model, via the introduction of suitable kernels propagating the collective activity of a neural population in time. Another approach is based on the use of large deviation techniques to study large networks of neurons [62]. This approach is inspired by the work on spin-glass dynamics, e.g. [63]. It takes into account the randomness of the maximum conductances and the noise at various levels. The individual neuron models are rate models, hence already mean-field models. The mean-field equations are not rigorously derived from the network equations in the limit of an infinite number of neurons, but they are shown to have a unique, non-Markov solution, i.e. with infinite memory, for each initial condition.

Brunel and Hakim considered a network of integrate-and-fire neurons connected with constant maximum conductances [41]. In the case of sparse connectivity, stationarity, and in a regime where individual neurons emit spikes at a low rate, they were able to analytically study the dynamics of the network and to show that it exhibits a sharp transition between a stationary regime and a regime of fast collective oscillations weakly synchronized. Their approach was based on a perturbative analysis of the Fokker-Planck equation. A similar formalism was used in [44] which, when complemented with self-consistency equations, resulted in the dynamical description of the mean-field equations of the network and was extended to a multi population network. Finally, Chizhov and Graham [64] have recently proposed a new method based on a population density approach allowing to characterize the mesoscopic behavior of neuron populations in conductance-based models.

Let us finish this very short and incomplete survey by mentioning the work of Sompolinsky and colleagues. Assuming a linear intrinsic dynamics for the individual neurons described by a rate model and random centered maximum conductances for the connections, they showed [65, 66] that the system undergoes a phase transition between two different stationary regimes: a ‘trivial’ regime where the system has a unique null and uncorrelated solution, and a ‘chaotic’ regime in which the firing rate converges towards a non-zero value and correlations stabilize on a specific curve which they were able to characterize.

All these approaches have in common that they are not based on the most widely accepted microscopic dynamics (such as the ones represented by the Hodgkin-Huxley equations or some of their simplifications) and/or involve approximations or moment closures. Our approach is distinct in that it aims at deriving rigorously and without approximations the mean-field equations of populations of neurons whose individual neurons are described by biological, if not correct at least plausible, representations. The price to pay is the complexity of the resulting mean-field equations. The specific study of their solutions is therefore a crucial step, which will be developed in forthcoming papers.