The analysis of dissipative, autonomous¹ dynamic flows (especially high-dimensional ones) can be greatly simplified, if a function can be constructed to provide an “energy landscape” to the problem. Note that only in very few cases can a nonlinear dynamical system be analytically solved; for instance, if the system itself is a quadratic form, one can use the Wei–Norman (Lie-algebraic) method to reduce it to a linear one (see e.g.,[1]). Energy landscapes not only help visualize the systems' phase space and its structural changes as parameters are varied, but allow predicting the rates of activated processes [2–4]. Some fields that benefit from the energy landscape approach are optimization problems [5], neural networks [6], protein folding [7], cell nets [8], gene regulatory networks [9, 10], ecology [11], and evolution [12].

For continuous-time flows, the possibility of this “Lyapunov function”—with its distinctive property Φ·<0 outside the attractors—was suggested in the context of the general stability problem of dynamical systems [13] and in a sense, it adds a quantitative dimension to the qualitative theory of differential equations. The linearization of the flow around its attractors always provides such a function, but its validity breaks down well inside their own basin. Instead, finding a global Lyapunov function is not an easy problem². If only the information of the (deterministic) dynamical system is to be used, this function can be found for the so-called “gradient flows”—purely irrotational flows in 3D, exact (longitudinal) forms in any dimensionality. But since for general relaxational flows (having nontrivial Helmholtz–Hodge decomposition) the integrability conditions are not automatically met, some more information is needed.

A hint of what information is needed comes from recalling that dynamical systems are models and as such, they leave aside a multitude of degrees of freedom—deemed irrelevant to the model, but nonetheless coupled to the “system.” A useful framework to deal with them is the one set forward by Langevin [15], which makes the dynamical flow into a stochastic process (thus nonautonomous, albeit driven by a stationary “white noise” process).

What Graham and his collaborators [16, 17] realized more than 30 years ago is that even in the deterministic limit, this space enlargement can eventually help meet the integrability conditions. Given an initial state x_i of a (continuous-time, dissipative, autonomous) dynamic flow ẋ = f (x), its conditional probability density function (pdf) P(x, t|x_i, 0) when submitted to a (Gaussian, centered) white noise ξ(t) with variance γ, namely³

obeys the Fokker–Planck equation (FPE)

in terms of the “drift” D⁽¹⁾ = f (x) and “diffusion” D⁽²⁾ = γ Kramers–Moyal coefficients [18–20]. Being the flow nonautonomous but dissipative, one can expect generically situations of statistical energy balance in which the pdf becomes stationary, ∂_tP_st(x) = 0, thus independent of the initial state. Then by defining Φ(x):=−∫xx0f(y)dy, it is immediate to find Pst(x)=N(x0)exp[-Φ(x)/γ].

For an n–component dynamic flow submitted to m ≤ n (Gaussian, uncorrelated, centered) white noises ξ_i(t) with common variance γ,

(σ is an n × m constant matrix) the nonequilibrium potential (NEP) has been thus defined [16] as

That implies Pst(x;γ)∝exp[-Φ(x;γ)/γ]+O(γ), which replaced into the stationary n–variable FPE ∇·[f(x)P_st(x) − γQ∇P_st(x)] = 0 (with Q : = σσ^T = Q^T) yields in the limγ→0 the equation

from which Φ(x) can in principle be found. In an attractor's basin, asymptotic stability imposes D : = det Q > 0. In fact, for m = n (restriction adopted hereafter) it is D = (det σ )², which in turn requires σ to be nonsingular. Using Equation (2),

for γ → 0. Hence, Φ(x) is a Lyapunov function for the deterministic dynamics.

Equation (2) has the structure of a Hamilton–Jacobi equation. This trouble can be circumvented if we can decompose f(x) = d(x) + r(x), with d(x) : = −Q∇Φ the dissipative part of f(x)⁴. Then Equation (2) reads r^T(x)∇Φ = 0, and r(x) is the conservative part of f(x). Note that d(x) is still irrotational (in the sense of the Helmholtz decomposition) but is not an exact form (the Hodge decomposition is made in the enlarged space).

For n = 2 we may write r(x) = κ Ω∇Φ, with Ω the N = 1 symplectic matrix. Hence f(x) = −(Q − κΩ)∇Φ, with det(Q − κΩ) = D + κ² > 0, and thus

For arbitrary real σ_ij we can parameterize

and define λ:=λ1λ2cos(α1-α2) (note that the condition D > 0 imposes α₂ ≠ α₁). Then

and Equation (3) reads

(∂_k is a shorthand for ∂/∂x_k). If a set {λ₁, λ₂, λ, κ} can be found such that Φ(x) fulfills the integrability condition ∂₂∂₁Φ = ∂₁∂₂Φ, then a NEP exists. Early successful examples are the complex Ginzburg–Landau equation (CGLE) [21, 22] and the FitzHugh–Nagumo (FHN) model [23, 24]⁵. This scheme has been later reformulated [29], extended [30], and exploited in many interesting cases [6–12].

The goal of this work is to show that a NEP exists for a broad class of rate models of neural networks, of the type proposed by Wilson and Cowan [31]. The Wilson–Cowan model has been used to model many different dynamics, brain areas, and neural-network structures in the brain. Therefore, the derivation of a NEP has potential implications for many problems in computational neuroscience. Section 2 is devoted to an analysis of the model and variations of section 3, to the derivation of the NEP in some of the cases studied in section 2, which are of high relevance in neuroscience. Section 4 undertakes a thorough discussion of our findings, and section 5 collects our conclusions.

Elucidating the architecture and dynamics of the neocortex is of utmost importance in neuroscience. But despite ongoing titanic efforts like the Human Brain Project or the BRAIN initiative, we are still very far from that goal. Given that the dynamics of single typical neurons has been relatively well described (in some cases even by analytical means), a fundamental approach can be practiced for small neural circuits. This means describing them as networks of excitable elements (neurons) connected by links (synapses), and solving the network dynamics by hybrid (analytical-numerical) techniques. However, the time employed in the analytical solution has poor scaling with size. Hence, this approach becomes unworkable for more than a few recurrently interconnected neurons, and one has to rely only on numerical simulations.

Fortunately—as evidenced since long ago by the existence (as in a medium) of wavelike excitations—the huge connectivity of the neocortex enables coarse-grained or mean-field descriptions, which provide more concise and relevant information to understand the mesoscopic dynamics of the system⁶. Frequently obtained via mean-field techniques and commonly referred to as rate models or neural mass models, coarse-grained reductions have been widely used in the theoretical study of neural systems [31, 33–35]. In particular, the one proposed by Wilson and Cowan [31] has proved to be very useful in describing the macroscopic dynamics of neural circuits. This level of description is able to capture many of the dynamical features associated with several cognitive and behavioral phenomena, such as working memory [33, 36] or perceptual decision making [35, 37]. It is also possible to use a rate model approach to study the dynamics of networks constituted by heterogeneous neurons [38, 39], thus recovering part of the complexity lost in the averaging. Disposing of an “energy function” (not restricted to symmetric couplings) for rate-level dynamics of neural networks would be a major added advantage.

The Wilson–Cowan model describes the evolution of competing populations x₁, x₂ of excitatory and inhibitory neurons, respectively. The model is defined by [31]

where x₁ and x₂ represent the coarse-grained activity of an excitatory and an inhibitory neural population, respectively, and the monotonically increasing (sigmoidal) response functions

are such that s_k(0) = 0, and range from -[1+exp(βkik0)]-1 for i_k → −∞ to 1-[1+exp(βkik0)]-1 for i_k → ∞. So the first crucial observation about the model is that it is asymptotically linear.

The currents i_k are in turn linearly related to the x_k:

All the parameters are real and moreover, the j_kl are positive (j₁₁ and j₂₂ are recurrent interactions, j₁₂ and j₂₁ are cross-population interactions). The above definitions are such that for M = 0, x = 0 is a stable fixed point. To avoid confusions in the following, note that det J = −(j₁₁j₂₂ − j₁₂j₂₁).

Wilson and Cowan [31] found interesting features as e.g., staircases of bistable regimes and limit cycles. A thorough analysis of the model's bifurcation structure has been undertaken in Borisyuk and Kirillov [40]. The authors create a two-parameter structural portrait by fixing all the parameters but μ₁ and j₂₁ and find that the μ₁−j₂₁ plane turns out to be partitioned into several regions by:

The uncoupled case (j₁₂ = j₂₁ = 0) is clearly a gradient system, with potential

where i₁ = j₁₁x₁ + μ₁ and i₂ = −j₂₂x₂ + μ₂. Functions F_k differ only in the values of their parameters. Their functional expression, involving polylogs, is uninteresting besides being complicated. Much more insight is obtained by observing the global features:

So it seems interesting to look at them in the limit β_k → ∞ (k = 1, 2),

θ(x) being Heaviside's unit step function. Unfortunately, neither Equation (4) nor their singular limit fulfill the above mentioned integrability condition.

In practice however, the names of Wilson and Cowan are associated to the broader class of rate models. In the following we shall show that the model defined by

does admit a NEP—for any functional forms of the nonlinear single-variable functions s_k(i_k)⁷—provided global stability is assured.

For the model defined by Equation (6), it is f(x) = (1τ1[-x1+s1(i1)]1τ2[-x2+s2(i2)]). The condition ∂₂∂₁Φ = ∂₁∂₂Φ, namely

boils down to

(and these in turn to j₂₁j₁₁τ₁λ₁ = j₁₂j₂₂τ₂λ₂) so that λ₂, λ and κ can be expressed in terms of λ₁, which sets the global scale of Φ(x):

Since r(x) = κ Ω∇Φ, τ₁ = τ₂ suffices to render the flow purely dissipative (albeit not gradient). From this, a good choice is λ1:=j22j21τ2ρ. In summary,

and Equation (3) becomes

Integrating Equation (7) over any path from x = 0, yields

The first—crucial—observation is that being s_k(i_k) sigmoidal functions, Equation (8) is at most quadratic. Global stability thus imposes detJ < 0, i.e., j₁₁j₂₂ > j₁₂j₂₁. But note that matrix J also determines the paraboloid's cross section⁸.

If the form xTAx, A=(j11j21−j12j21−j12j21j12j22) in the first term of Equation (8) could be straightforwardly factored out, then one could tell what its section is by watching whether the factors are real or complex. A more systematic approach is to reduce x^TAx to canonical form by a similarity transformation that involves the normalized eigenvectors of A. Then the inverse squared lengths of the principal axes are the eigenvalues λ1,2=12(j11j21+j12j22)±14(j11j21+j12j22)2+j12j21detJ of A. Since the j_kl are positive and detJ < 0, the second term is lesser than the first and the cross section is definitely elliptic. Although global stability rules out detJ > 0, we can conclude that the instability proceeds through a pitchfork (codimension one) bifurcation along the minor principal axis (because of the double role of detJ), not a Hopf (codimension two) one.

For the remaining terms, we note that Equation (7) can be written as ∇Φ=1ρτ1τ2detJ(j2100-j12)J(x-s)and recall that s_k(i_k) have sigmoidal shape. So at large |x|, the component -1ρτ1τ2detJ(j2100-j12)Js will tend to different constants—according to the signs of i_k—so the asymptotic contribution of these terms will be piecewise linear, namely a collection of half planes.

The reduction to the uncoupled case can be safely done by writing j₁₂ = ϵ and j₂₁ = αϵ:

To first order as ϵ → 0, one retrieves

so by choosing α=τ2j22τ1j11 and ρ=ϵτ1j11,

A popular choice—that can be cast in the form of Equation (5)—is sk(ik):=νk2(1+tanhβkik), β_k > 0, for which⁹

Its β_k → ∞ limit, ν_kθ(i_k) with

highlights the cores of the response functions while keeping the global landscape¹⁰.

As a check of Equation (8), we show in the next subsections the mechanism whereby Equation (6) can sustain bistability.

When τ₁ ≠ τ₂, then

and d(x) = f(x)−r(x). However Φ(x) can remain the same, as far as τ₁ τ ₂ does not change. So whereas the contour plots of the NEP in Figure 3 reproduce those of Figures 2C,D, the displayed set of trajectories (from random initial conditions within suitably selected tiny patches) have r(x)≠0 and consequently, many of them perform a large excursion toward the attractor.

Excitable events such as those described here by the NEP, in which the activity of excitatory neurons in the population shows a sharp peak, are known in the computational neuroscience literature as “population bursts.” These are brief events of high excitatory activity in the neural system being modeled. In neural network models composed of interconnected spiking neurons, they reflect a sudden rise in spiking activity at the level of the whole population (or a significant part of it), in such a way that a high proportion of neurons in the network fire at least one action potential during a short time window. Spiking neurons participating in the population burst are therefore transiently synchronized. In spite of not being able to properly capture synchronous phenomena, the Wilson–Cowan model may capture this phenomenon as a transient peak of activity that is later shut down by inhibition. But for more realistic models, additional biophysical mechanisms (such as actual spiking dynamics, refractory period of neurons or short-term adaptation) have to be considered since they are likely involved in population bursts on real neural systems [41]. Population bursts have several computational uses; for example, they can be used to transmit temporally precise information to other brain areas, even in the presence of noise or heterogeneity [38].

Rate- (also called neural mass-) models have been a useful approach to neural networks for half a century. Today, their simplicity (not short of comprehensivity) makes them ideal to fulfill the node dynamics in, for instance, connectome-based brain networks. So the availability of an “energy function” for rate models is expected to be welcome news.

Dynamical systems of the form given by Equation (6) admit a NEP regardless of the functional forms of the nonlinear single-variable functions s_k(i_k). Throughout this work, the latter are assumed to have the same functional form, of sigmoidal shape. But neither condition is necessary to satisfy the integrability condition.

A crucial observation about rate models—even the one put forward by Wilson and Cowan [31], and given by Equations (4)–(5)—is that they are asymptotically linear, so their eventual NEP can be at most quadratic. Then in principle, global stability rules out some coupling configurations. Obviously, this requirement can be relaxed if the rate model fulfills the node dynamics of a neural network, for what matters in that case is the network's global stability.

The here obtained NEP provides a more quantitative intuition on the phenomenon of bistability, that has been naturally found in real neural systems. Neural bistability underlies e.g., the persistent activity which is commonly found in neurons of the prefrontal cortex, a mechanism that is thought to maintain information during working memory tasks [36, 42]. In the presence of neural noise and other adaptation mechanisms, bistability is also a useful hypothesis to explain slow irregular dynamics or “up” and “down” dynamics, also observed across cortex and modeled using bistable dynamics [43–45].

Our results open the door to considering the calculation of nonequilibrium potentials of rate-based neural network models, and in particular considering the implications of different biologically realistic dynamics in such potentials. One interesting possibility is to consider in our model the effect of short-term synaptic plasticity effects. Short-term plasticity has been shown to impact computational properties of neural systems, such as their signal detection abilities [46–48], their pattern storage capacity [49, 50], or the statistics of neural bistable dynamics [44, 45]. We would expect, for example, that changes in the ability of neural systems to detect weak signals due to short-term plasticity could be reflected in changes in the nonequilibrium potential landscape, making population bursts easier to be triggered by weak stimuli. Changes in the statistics of neural bistable dynamics, or ‘up-down' transitions, could be reflected in swifts of the dwells in the landscape, and also on the statistics of real experimental data.

Finally, it is worth mentioning that the here obtained NEP—valid as argued for generic transfer functions s_k(i_k)—opens the door to the potential use of more generic rate models in the field of artificial neural networks and deep learning. By identifying the NEP with the cost function to be minimized, gradient descent algorithms can be used to train networks of generic Wilson–Cowan units for different tasks. This implies that more realistic and less computationally expensive neural population models can be trained and used for behavioral tasks, a topic that has gathered attention recently [51, 52].

All the listed authors have made substantial and direct intellectual contribution to this work, and approve its publication. In particular, JM proposed the idea, contributed preliminary analytical calculations, and provided key information from the field of neuroscience.