In the deterministic limit (\u03f5,\sigma \to 0) the neural Langevin equation (16) reduces to the mean-field equation (6). Suppose that the latter supports a stable limit cycle solution of the form \overline{\mathbf{x}}={\mathbf{x}}^{\ast}(t) with {\mathbf{x}}^{\ast}(t+nT)={\mathbf{x}}^{\ast}(t) for all integers *t*, where *T* is the period of the oscillations. The Langevin equation (16) then describes a noise-driven population oscillator. Now consider an ensemble of \mathcal{N} identical population oscillators each of which consists of *M* interacting sub-populations evolving according to a Langevin equation of the form (16). We ignore any coupling between different population oscillators, but assume that all oscillators are driven by a common source of extrinsic noise. Introducing the ensemble label *μ*\mu =1,\dots ,\mathcal{N}, we thus have the system of Langevin equations

\begin{array}{rl}d{X}_{k}^{(\mu )}=& {A}_{k}\left({\mathbf{X}}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}dt+\u03f5{b}_{k}\left({\mathbf{X}}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}d{W}_{k}^{(\mu )}(t)\\ +\sigma {a}_{k}\left({\mathbf{X}}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}dW(t),\phantom{\rule{1em}{0ex}}k=1,\dots ,M.\end{array}

(18)

We associate an independent set of Wiener processes {W}_{k}^{(\mu )}k=1,\dots ,M, with each population oscillator (independent noise) but take the extrinsic noise to be given by a single Wiener process W(t) (common noise):

\u3008d{W}_{k}^{(\mu )}(t)\phantom{\rule{0.2em}{0ex}}d{W}_{l}^{(\nu )}(t)\u3009={\delta}_{k,l}{\delta}_{\mu ,\nu}\phantom{\rule{0.2em}{0ex}}dt,

(19)

\u3008d{W}_{k}^{(\mu )}(t)\phantom{\rule{0.2em}{0ex}}dW(t)\u3009=0,

(20)

\u3008dW(t)\phantom{\rule{0.2em}{0ex}}dW(t)\u3009=dt.

(21)

Langevin equations of the form (18) have been the starting point for a number of recent studies of noise-induced synchronization of uncoupled limit cycle oscillators [9, 11–15]. The one major difference from our own work is that these studies have mostly been motivated by single neuron oscillator models, in which both the independent and common noise sources are extrinsic to the oscillator. In contrast, we consider a stochastic population model in which the independent noise sources are due to finite size effects intrinsic to each oscillator. The reduction of the neural master equation (1) to a corresponding Langevin equation (16) then leads to multiplicative rather than additive noise terms; this is true for both intrinsic and extrinsic noise sources. We will show that this has non-trivial consequences for the noise-induced synchronization of an ensemble of population oscillators. In order to proceed, we carry out a stochastic phase reduction of the full Langevin equations (18), following the approach of Nakao *et al.* [12] and Ly and Ermentrout [15]. We will only sketch the analysis here, since further details can be found in these references. We do highlight one subtle difference, however, associated with the fact that the intrinsic noise terms are Ito rather than Stratonovich.

### 3.1 Stochastic phase reduction

Introduce the phase variable \theta \in (-\pi ,\pi ] such that the dynamics of an individual limit cycle oscillator (in the absence of noise) reduces to the simple phase equation \dot{\theta}=\omega, where \omega =2\pi /T is the natural frequency of the oscillator and \overline{\mathbf{x}}(t)={\mathbf{x}}^{\ast}(\theta (t)). The phase reduction method [6, 7] exploits the observation that the notion of phase can be extended into a neighborhood \mathcal{M}\subset {\mathbb{R}}^{2} of each deterministic limit cycle, that is, there exists an isochronal mapping \Psi :\mathcal{M}\to [-\pi ,\pi ) with \theta =\Psi (\mathbf{x}). This allows us to define a stochastic phase variable according to {\Theta}^{(\mu )}(t)=\Psi ({\mathbf{X}}^{(\mu )}(t))\in [-\pi ,\pi ) with {\mathbf{X}}^{(\mu )}(t) evolving according to equation (18). Since the phase reduction method requires the application of standard rules of calculus, it is first necessary to convert the intrinsic noise term in equation (18) to a Stratonovich form [25, 49]:

\begin{array}{rl}d{X}_{k}^{(\mu )}=& [{A}_{k}\left({\mathbf{X}}^{(\mu )}\right)-\frac{{\u03f5}^{2}}{2}{b}_{k}\left({\mathbf{X}}^{(\mu )}\right)\frac{\partial {b}_{k}({\mathbf{X}}^{(\mu )})}{\partial {X}_{k}^{(\mu )}}]\phantom{\rule{0.2em}{0ex}}dt\\ +\u03f5{b}_{k}\left({\mathbf{X}}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}d{W}_{k}^{(\mu )}(t)+\sigma {a}_{k}\left({\mathbf{X}}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}dW(t).\end{array}

(22)

The phase reduction method then leads to the following Stratonovich Langevin equations for the the stochastic phase variables {\Theta}^{(\mu )}\mu =1,\dots ,\mathcal{N} [9, 12, 14]:

Here {Z}_{k}(\theta ) is the *k* th component of the infinitesimal phase resetting curve (PRC) defined as

{Z}_{k}(\theta )=\frac{\partial \Psi (\mathbf{x})}{\partial {x}_{k}}{|}_{\mathbf{x}={\mathbf{x}}^{\ast}(\theta )}

(24)

with {\sum}_{k=1}^{M}{Z}_{k}(\theta ){A}_{k}({\mathbf{x}}^{\ast}(\theta ))=\omega. All the terms multiplying {Z}_{k}(\theta ) are evaluated on the limit cycle so that

\begin{array}{rl}{a}_{k}(\theta )& ={a}_{k}({\mathbf{x}}^{\ast}(\theta )),\phantom{\rule{1em}{0ex}}{b}_{k}(\theta )={b}_{k}({\mathbf{x}}^{\ast}(\theta )),\\ \partial {b}_{k}(\theta )& =\frac{\partial {b}_{k}(\mathbf{x})}{\partial {x}_{k}}{|}_{{\mathbf{x}}_{k}={\mathbf{x}}_{k}^{\ast}(\theta )}.\end{array}

(25)

It can be shown that the PRC is the unique 2*π*-periodic solution of the adjoint linear equation [7]

\frac{d{Z}_{k}}{dt}=-\sum _{l=1}^{M}{A}_{lk}({\mathbf{x}}^{\ast}(t)){Z}_{l}(t),

(26)

where {A}_{lk}=\partial {A}_{l}/\partial {x}_{k}, which is supplemented by the normalization condition {\sum}_{k}{Z}_{k}(t)\phantom{\rule{0.2em}{0ex}}d{x}_{k}^{\ast}/dt=\omega. The PRC can thus be evaluated numerically by solving the adjoint equation backwards in time. (This exploits the fact that all non-zero Floquet exponents of solutions to the adjoint equation are positive.) It is convenient to rewrite equation (23) in the more compact form

\begin{array}{rl}d{\Theta}^{(\mu )}=& [\omega -\frac{{\u03f5}^{2}}{2}\Omega \left({\Theta}^{(\mu )}\right)]\phantom{\rule{0.2em}{0ex}}dt\\ +\u03f5\sum _{k=1}^{M}{\beta}_{k}\left({\Theta}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}d{W}_{k}^{(\mu )}(t)+\sigma \alpha \left({\Theta}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}dW(t),\end{array}

(27)

where

\begin{array}{rl}{\beta}_{k}(\theta )& ={Z}_{k}(\theta ){b}_{k}(\theta ),\phantom{\rule{1em}{0ex}}\Omega (\theta )=\sum _{k=1}^{M}{Z}_{k}(\theta ){b}_{k}(\theta )\phantom{\rule{0.2em}{0ex}}\partial {b}_{k}(\theta ),\\ \alpha (\theta )& =\sum _{k=1}^{M}{Z}_{k}(\theta ){a}_{k}(\theta ).\end{array}

(28)

In order to simplify the analysis of noise-induced synchronization, we now convert equation (27) from a Stratonovich to an Ito system of Langevin equations:

d{\Theta}^{(\mu )}={\mathcal{A}}^{(\mu )}(\mathbf{\Theta})\phantom{\rule{0.2em}{0ex}}dt+d{\zeta}^{(\mu )}(\mathbf{\Theta},t),

(29)

where \{{\zeta}^{(\mu )}(\mathbf{\Theta},t)\} are correlated Wiener processes with \mathbf{\Theta}=({\Theta}^{(1)},\dots ,{\Theta}^{(\mathcal{N})}). That is,

d{\zeta}^{(\mu )}(\mathbf{\Theta},t)=\u03f5\sum _{k=1}^{M}{\beta}_{k}\left({\Theta}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}d{W}_{k}^{(\mu )}(t)+\sigma \alpha \left({\Theta}^{(\mu )}\right)\phantom{\rule{0.2em}{0ex}}dW(t),

(30)

with \u3008d{\zeta}^{(\mu )}(\mathbf{\Theta},t)\u3009=0 and \u3008d{\zeta}^{(\mu )}(\mathbf{\Theta},t)\phantom{\rule{0.2em}{0ex}}d{\zeta}^{(\nu )}(\mathbf{\Theta},t)\u3009={C}^{(\mu \nu )}(\mathbf{\Theta})\phantom{\rule{0.2em}{0ex}}dt, where {C}^{(\mu \nu )}(\mathit{\theta}) is the equal-time correlation matrix

{C}^{(\mu \nu )}(\mathit{\theta})={\sigma}^{2}\alpha \left({\theta}^{(\mu )}\right)\alpha \left({\theta}^{(\nu )}\right)+{\u03f5}^{2}\sum _{k=1}^{M}{\beta}_{k}\left({\theta}^{(\mu )}\right){\beta}_{k}\left({\theta}^{(\nu )}\right){\delta}_{\mu ,\nu}.

(31)

The drift term {\mathcal{A}}^{(\mu )}(\mathit{\theta}) is given by

{\mathcal{A}}^{(\mu )}(\mathit{\theta})=\omega -\frac{{\u03f5}^{2}}{2}\Omega \left({\theta}^{(\mu )}\right)+\frac{1}{4}{\mathcal{B}}^{\prime}\left({\theta}^{(\mu )}\right)

(32)

with

\begin{array}{rl}\mathcal{B}\left({\theta}^{(\mu )}\right)& \equiv {C}^{(\mu \mu )}(\mathit{\theta})\\ ={\sigma}^{2}{\left[\alpha \left({\theta}^{(\mu )}\right)\right]}^{2}+{\u03f5}^{2}\sum _{k=1}^{M}{\left[{\beta}_{k}\left({\theta}^{(\mu )}\right)\right]}^{2}.\end{array}

(33)

It follows that the ensemble is described by a multivariate Fokker-Planck equation of the form

\begin{array}{rl}\frac{\partial P(\mathit{\theta},t)}{\partial t}=& -\sum _{\mu =1}^{\mathcal{N}}\frac{\partial}{\partial {\theta}^{\mu}}[{\mathcal{A}}^{(\mu )}(\mathit{\theta})P(\mathit{\theta},t)]\\ +\frac{1}{2}\sum _{\mu ,\nu =1}^{\mathcal{N}}\frac{{\partial}^{2}}{\partial {\theta}^{\mu}\phantom{\rule{0.2em}{0ex}}\partial {\theta}^{\nu}}[{C}^{(\mu \nu )}(\mathit{\theta})P(\mathit{\theta},t)].\end{array}

(34)

Equation (34) was previously derived by Nakao *et al.* [12] (see also [15]). Here, however, there is an additional contribution to the drift term {\mathcal{A}}^{(\mu )} arising from the fact that the independent noise terms appearing in the full system of Langevin equations (18) are Ito rather than Stratonovich, reflecting the fact that they arise from finite size effects.

### 3.2 Steady-state distribution for a pair of oscillators

Having obtained the FP equation (34), we can now carry out the averaging procedure of Nakao *et al.* [12]. The basic idea is to introduce the slow phase variables \mathit{\psi}=({\psi}^{(1)},\dots ,{\psi}^{(\mathcal{N})}) according to {\theta}^{\mu}=\omega t+{\psi}^{\mu} and set Q(\mathit{\psi},t)=P(\{\omega t+{\theta}^{(\mu )}\},t). For sufficiently small *ϵ* and *σ* *Q* is a slowly varying function of time so that we can average the Fokker-Planck equation for *Q* over one cycle of length T=2\pi /\omega. The averaged FP equation for *Q* is thus [12]

\begin{array}{rl}\frac{\partial Q(\mathit{\psi},t)}{\partial t}=& \frac{{\u03f5}^{2}}{2}\overline{\Omega}\sum _{\mu =1}^{\mathcal{N}}\frac{\partial}{\partial {\psi}^{\mu}}Q(\mathit{\psi},t)\\ +\frac{1}{2}\sum _{\mu ,\nu =1}^{\mathcal{N}}\frac{{\partial}^{2}}{\partial {\psi}^{\mu}\phantom{\rule{0.2em}{0ex}}\partial {\psi}^{\nu}}[{\overline{C}}^{(\mu \nu )}(\mathit{\psi})Q(\mathit{\psi},t)],\end{array}

(35)

where

\overline{\Omega}=\frac{1}{2\pi}{\int}_{0}^{2\pi}\Omega (\theta )\phantom{\rule{0.2em}{0ex}}d\theta ,

(36)

and

{\overline{C}}^{(\mu \nu )}(\mathit{\psi})={\sigma}^{2}g({\psi}^{(\mu )}-{\psi}^{(\nu )})+{\u03f5}^{2}h(0){\delta}_{\mu ,\nu}

(37)

with

\begin{array}{rl}g(\psi )& =\frac{1}{2\pi}{\int}_{-\pi}^{\pi}\alpha ({\theta}^{\prime})\alpha ({\theta}^{\prime}+\psi )\phantom{\rule{0.2em}{0ex}}d{\theta}^{\prime},\\ h(\psi )& =\frac{1}{2\pi}{\int}_{-\pi}^{\pi}\sum _{k=1}^{M}{\beta}_{k}({\theta}^{\prime}){\beta}_{k}({\theta}^{\prime}+\psi )\phantom{\rule{0.2em}{0ex}}d{\theta}^{\prime}.\end{array}

(38)

Following Nakao *et al.* [12] and Ly and Ermentrout [15], we can now investigate the role of finite size effects on the noise-induced synchronization of population oscillators by focussing on the phase difference between two oscillators. Setting \mathcal{N}=2 in equation (35) gives

\begin{array}{rcl}\frac{\partial Q}{\partial t}& =& \frac{{\u03f5}^{2}}{2}\overline{\Omega}[\frac{\partial Q}{\partial {\psi}^{(1)}}+\frac{\partial Q}{\partial {\psi}^{(2)}}]+\frac{1}{2}[{\sigma}^{2}g(0)+{\u03f5}^{2}h(0)]\\ \times \left[\right(\frac{\partial}{\partial {\psi}^{(1)}}{)}^{2}+{\left(\frac{\partial}{\partial {\psi}^{(2)}}\right)}^{2}]Q\\ +\frac{{\partial}^{2}}{\partial {\psi}^{(1)}\phantom{\rule{0.2em}{0ex}}\partial {\psi}^{(2)}}\left[{\sigma}^{2}g\right({\psi}^{(1)}-{\psi}^{(2)}\left)Q\right].\end{array}

Performing the change of variables

\psi =({\psi}^{(1)}+{\psi}^{(1)})/2,\phantom{\rule{1em}{0ex}}\varphi ={\psi}^{(1)}-{\psi}^{(1)}

and writing Q({\psi}^{(1)},{\psi}^{(2)},t)=\Psi (\psi ,t)\Phi (\varphi ,t) we obtain the pair of PDEs

\frac{\partial \Psi}{\partial t}=\frac{{\u03f5}^{2}}{2}\overline{\Omega}\frac{\partial \Psi}{\partial \psi}+\frac{1}{4}\left[{\sigma}^{2}\right(g(0)+g(\varphi ))+{\u03f5}^{2}h(0)]\frac{{\partial}^{2}\Psi}{\partial {\psi}^{2}}

and

\frac{\partial \Phi}{\partial t}=\frac{{\partial}^{2}}{\partial {\varphi}^{2}}\left[{\sigma}^{2}\right(g(0)-g(\varphi ))+{\u03f5}^{2}h(0)]\Phi .

These have the steady-state solution

{\Psi}_{0}(\psi )=\frac{1}{2\pi},\phantom{\rule{1em}{0ex}}{\Phi}_{0}(\varphi )=\frac{{\Gamma}_{0}}{{\sigma}^{2}(g(0)-g(\varphi ))+{\u03f5}^{2}h(0)},

(39)

where Γ_{0} is a normalization constant.

A number of general results regarding finite size effects immediately follow from the form of the steady-state distribution {\Phi}_{0}(\varphi ) for the phase difference *ϕ* of two population oscillators. First, in the absence of a common extrinsic noise source (\sigma =0) and \u03f5>0{\Phi}_{0}(\varphi ) is a uniform distribution, which means that the oscillators are completely desynchronized. On the other hand, in the thermodynamic limit N\to \infty we have \u03f5={N}^{-1/2}\to 0 so that the independent noise source vanishes. The distribution {\Phi}_{0}(\varphi ) then diverges at \theta =0 while keeping positive since it can be shown that g(0)\ge g(\theta ) [12]. Hence, the phase difference between any pair of oscillators accumulates at zero, resulting in complete noise-induced synchronization. For finite *N*, intrinsic noise broadens the distribution of phase differences. Taylor expanding g(\varphi ) to second order in *ϕ* shows that, in a neighbourhood of the maximum at \varphi =0, we can approximate {\Phi}_{0}(\varphi ) by the Cauchy distribution

{\Phi}_{0}(\varphi )\approx \frac{{\Gamma}_{0}^{\prime}}{{\varphi}^{2}{\sigma}^{2}|{g}^{\u2033}(0)|/2+h(0)/N}

for an appropriate normalization {\Gamma}_{0}^{\prime}. Thus the degree of broadening depends on the ratio

\Delta =\frac{h(0)}{N{\sigma}^{2}|{g}^{\u2033}(0)|}.

The second general result is that the functions \alpha (\theta ) and {\beta}_{k}(\theta ) that determine g(\varphi ) and h(\varphi ) according to equations (38) are nontrivial products of the phase resetting curves {Z}_{k}(\theta ) and terms {a}_{k}(\theta ){b}_{k}(\theta ) that depend on the transition rates of the original master equation, see equations (17), (25) and (28). This reflects the fact that both intrinsic and extrinsic noise sources in the full neural Langevin equation (18) are multiplicative rather than additive. As previously highlighted by Nakao *et al.* [12] for a Fitzhugh-Nagumo model of a single neuron oscillator, multiplicative noise can lead to additional peaks in the function g(\varphi ), which can induce clustering behavior within an ensemble of noise-driven oscillators. In order to determine whether or not a similar phenomenon occurs in neural population models, it is necessary to consider specific examples. We will consider two canonical models of population oscillators, one based on interacting sub-populations of excitatory and inhibitory neurons and the other based on an excitatory network with synaptic depression.