In this paper, we have presented and demonstrated the use of a computationally efficient method for systematically investigating the effects of heterogeneity in the parameters of a coupled network of neural oscillators. The method constitutes a model reduction approach: By only considering oscillators with parameter values given by roots of families of orthogonal polynomials (Legendre, Hermite, …), we can use the Gaussian quadrature to accurately evaluate the term coupling the oscillators, which can be thought of as the discretisation of an integral over the heterogeneous dimension(s).

Effectively, we are simulating the behaviour of an infinite number of oscillators by only simulating a small number of judiciously selected ones, modifying appropriately the way they are coupled. When the oscillators are synchronised, or at a fixed point, the function to be integrated is a smooth function of the heterogeneous parameter(s), and thus, convergence is very rapid. The technique is general (although subject to the restriction immediately above) and can be used when there is more than one heterogeneous parameter, via full or sparse tensor products in parameter space. For a given level of accuracy, we are simulating far fewer neurons than might naively be expected. The emphasis here has been on computational efficiency rather than a detailed investigation of parameter dependence.

The model we considered involved coupling only through the mean of a function, *s*, of the variable ${V}_{i}$ which, in the limit $N\to \mathrm{\infty}$, can be thought of as an integral or, more generally, as a functional of $V(\mu )$. Thus, the techniques demonstrated here could also be applied to networks coupled through terms which, in the continuum limit, are integrals or functions of integrals. A simple example is diffusive coupling [3]; another possibility is coupling which is dependent upon the correlation between some or all of the variables. As mentioned, the technique will break down once the oscillators become desynchronised, as the dependence of state on parameter(s) will no longer be smooth. However, if the oscillators form several clusters [14, 36], it may be possible to apply the ideas presented here to each cluster, as the dependence of state on parameter(s) *within* each cluster should still be smooth. Ideally, this reparametrisation would be done adaptively as clusters form, in the same way that algorithms for numerical integration adapt as the solution varies [30]. Alternatively, if a single oscillator ‘breaks away’ [27], the methods presented here could be used on the remaining synchronous oscillators, with the variables describing the state of the rogue oscillator also fully resolved. More generally, there are systems in which it is not necessarily the *state* of an oscillator that is a smooth function of the heterogeneous parameter, but the *parameters describing the distribution of states*[37, 38], and the ideas presented here could also be useful in this case.

The primary study with which we should compare our results is that of Rubin and Terman [14]. They considered essentially the same model as Equations 1 and 2 but with heterogeneity only in the ${I}_{\mathrm{app}}$ and, taking the continuum limit, referred to the curve in $(V,h)$ space describing the state of the neurons at any instant in time as a ‘snake’. By making various assumptions, such as an infinite separation of time scales between the dynamics of the ${V}_{i}$ and the ${h}_{i}$, and that the dynamics of the ${h}_{i}$ in both the active and quiescent phases is linear, they derived an expression for the snake at one point in its periodic orbit and showed that such a snake is unique and stable. They also estimated the parameter values at which the snake ‘breaks’ and some oscillators lose synchrony. In contrast with their mainly analytical study, ours is mostly numerical and thus does not rely on any of the assumptions just mentioned. Using the techniques presented here, we were able to go beyond the work of Rubin and Terman, exploring parameter space.

Our approach can be thought of as a particular parametrisation of this snake, which takes into account the probability density of the heterogeneity parameter(s); we also showed a systematic way of extending this one-dimensional snake to two and higher dimensions. Another paper which uses some of the same ideas as presented here is that of Laing and Kevrekidis [3]. There, the authors considered a finite network of coupled oscillators and used a polynomial chaos expansion of the same form as Equation 12. However, instead of integrating the equations for the polynomial chaos coefficients directly, they used projective integration [39] to do so, in an ‘equation-free’ approach [40] in which the equations satisfied by the polynomial chaos coefficients are never actually derived. They also chose the heterogeneous parameter values randomly from a prescribed distribution and averaged over realisations of this process in order to obtain ‘typical’ results. Similar ideas had been explored earlier by Moon *et al.*[27], who considered a heterogeneous network of phase oscillators.

Assisi *et al.*[22] considered a heterogeneous network of coupled neural oscillators, deriving equations of similar functional form to Equations 9 and 11. Their approach was to expand the variables in a way similar to Equation 12 but using a small number of arbitrarily chosen ‘modes’ rather than orthogonal polynomials. Their choice of modes, along with the fact that their neural model consisted of ODEs with polynomial right hand sides, allowed them to analytically derive the ODEs satisfied by the coefficients of the modes. This approach allowed them to qualitatively reproduce some of the behaviour of the network such as the formation of two clusters of oscillators. However, in the general case modes should be chosen as orthogonal polynomials, the specific forms of which are determined by the distribution of the heterogeneous parameter(s) [25, 26].

The network we considered was all-to-all coupled, and the techniques presented should be applicable to other similar systems. The only requirement is that the relationship between the heterogeneity parameter(s) and the state of the system (possibly after transients) be smooth (or possibly piecewise smooth). An interesting extension is the case when the network under consideration is not all-to-all. Then, the effects of degree distribution may affect the dynamics of individual oscillators [38, 41, 42], and if we have a way of parameterising this type of heterogeneity, it might be possible to apply the ideas presented here to such networks. Degree distribution is a discrete variable, and corresponding families of orthogonal polynomials exist for a variety of discrete random variables [25, 26].