Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.
Modeling the brain is confounded by the fact that there are a very large number of neurons, and the neurons are heterogeneous and individually complex. Given current analytical and computational capabilities, we can either study neuronal dynamics in some biophysical detail for a small or medium set of neurons or consider a large population of abstract simplified neural units. We then can extrapolate only to the desired regime of large numbers of biophysical neurons. In particular, there is a dichotomy between network models that incorporate Hodgkin-Huxley or integrate-and-fire spiking dynamics and models that include only the rate or activity of neural units. While the rate description has yielded valuable insights into many neural phenomena, it cannot describe physiological phenomena thought to be important for neural processing such as synchronization, spike-time-dependent plasticity, or any correlated activity at the spike level. Likewise, it is difficult to analyze or simulate a large network of spiking neurons. Our goal is to derive an intermediate description of neural activity that is complex enough to account for spike-level correlations yet simple enough to be amenable to analysis and numerical computation for large networks.
Rate or activity equations have been a standard tool of computational and theoretical neuroscience, early important examples being the work of Wilson and Cowan, Cohen and Grossberg, Amari and Hopfield (Wilson & Cowan, 1972, 1973; Amari, 1975, 1977; Hopfield, 1984; Cohen & Grossberg, 1983). Models of this type have been used to investigate pattern formation, visual hallucinations, content-addressable memory, and many other topics (Ermentrout & Cowan, 1979; Hopfield, 1984; Ermentrout, 1998; Bressloff, Cowan, Golubitsky, Thomas, & Wiener, 2001; Coombes, 2005). Naturally these equations are so-called because they describe the evolution of a neural activity variable often ascribed to the firing rate or synaptic drive of a population of interacting neurons (Ermentrout, 1998; Gerstner, 2000). These equations are considered to represent the neural dynamics averaged over time or population of a more complicated underlying process. In general, these activity equations make an implicit assumption that correlated firing is unimportant. They are a mean field theory that captures the dynamics of the mean firing rate or activity that is independent of the influence of correlations, which in some cases may alter the dynamics considerably. As an example, the effects of synchrony, which have been proposed to be important for neural processing (Gray & Singer, 1989; Beshel, Kopell, & Kay, 2007), are not included. Here, we give a systematic prescription to extend rate models to account for these effects.
An analogy for our problem and approach can be made to the field of equilibrium statistical mechanics. The statistics of such systems (e.g., the Ising model) in thermal equilibrium are described by a partition function, an integration over all configurations available to the system. For the Ising model, this refers to all possible configurations of the individual spins. The partition function is akin to the generating function for a statistical distribution from which the moments or cumulants can be obtained. For the Ising model, the first moment corresponds to the mean magnetization, and the second moment describes the mean correlation between the spins. The linear response of the system is the magnetic susceptibility, which describes the reaction of the system to an external input. In general, the partition function cannot be summed or integrated explicitly. However, these moments can be obtained perturbatively by using the method of steepest descents to approximate the partition function. This then yields a systematic expansion, and the lowest order is called mean field theory, since all higher cumulants are zero. By computing the expansion to higher order, the effects of correlations and fluctuations can be included.
This procedure requires full knowledge of the underlying microscopic theory that is to be averaged over. In neuroscience, the underlying model is not completely known; it would require full knowledge of the different types of neurons, their membrane and synaptic kinetics, and their synaptic connectivity. However, given a particular mean field theory, one can ask about the minimal constraints this theory places on the microscopic theory and its asymptotic expansion. Thus, although the full microscopic theory cannot be reconstructed, by constraining the expansion, the mean field theory can dictate the minimal structure of any extension of a set of rate equations. In this letter, we consider a well-known neural rate equation and deduce the minimal structure we expect for a consistent extension that includes correlations.
Buice and Cowan (2007) previously adapted a path integral formalism used in nonequilibrium statistical mechanics (Doi, 1976a, 1976b; Peliti, 1985) to analyze the dynamics of a Markov model for neural firing. They derived a generating functional (expressed as an infinite-dimensional path integral), which is specified by an “action” for the complete dynamical distribution of the model, and showed that the mean field theory for that system corresponded to a Wilson-Cowan type of rate equation. They then analyzed the scaling properties for the correlations near criticality. They showed how mean field theory could be corrected by using steepest descents to generate a systematic expansion that describes the effect of correlations. Here, we show that a moment hierarchy can be constructed by taking explicit averages of the Markov model. Each equation in the moment hierarchy is coupled to higher moments in the hierarchy. The hierarchy can be made useful as a calculational tool for the statistics of the dynamics if it can be truncated. We show that the moment hierarchy and the generating functional are equivalent and that the equations of the hierarchy are the equations of motion of the action in the generating functional. The truncation condition for the perturbation series of the path integral is also a truncation condition for the hierarchy. This provides for both a compact description of network statistics and a natural truncation or closure condition for a moment hierarchy. We can also show using the path integral formalism that the Markov model is a natural minimal extension of the Wilson-Cowan rate equation.
Approaches to neural network modeling using statistical mechanics are not new (Cowan, 1991; Hopfield, 1982, 1984; Ohira & Cowan, 1993; Peretto, 1984; Amit, Gutfreund, & Sompolinsky, 1985). Those works were largely concerned with models adhering to detailed balance, whereas we make the explicit assumption that neural dynamics admits an absorbing state that violates detailed balance. In the absence of internal activity and external stimulation, there will be no activity in the network. Other studies using a stochastic description of neural dynamics have considered the neurons in a background of Poisson activity with disorder in the connectivity (Amit & Brunel, 1997a, 1997b), or considered neural activity as a renewal process (Gerstner, 1995, 2000; Gerstner & Kistler, 2002). Van Vreeswijk and Sompolinsky (1996, 1998) demonstrated that disorder in network activity can arise purely as a result of disorder in the connectivity, without stochastic input. Kinetic theory and density approaches are investigated in Nykamp and Tranchina (2000), Cai, Tao, Shelley, and McLaughlin (2004), Ly and Tranchina (2007), and mean field density approaches to the asynchronous state appear in Abbott and Van Vreeswijk (1993) and Treves (1993). Golomb and Hansel (2000) study synchrony in sparse networks via a reduction to a phase model. Fokker-Planck approaches for networks appear in Fusi and Mattia (1999), Brunel and Hakim (1999), Brunel (2000), and Brunel and Hansel (2006). Responses of single neurons driven by noise appear in Plesser and Gerstner (2000), Salinas and Sejnowski (2002), Fourcaud and Brunel (2002), and Soula, Beslon, and Mazet (2006). Approaches to correlated neural activity including finite size effects appear in Ginzburg and Sompolinsky (1994), Mattia and Del Giudice (2002), Soula and Chow (2007), and El Boustani and Destexhe (2009). El Boustani and Destexhe (2009) develop a moment hierarchy for a Markov model of asynchronous irregular states of neural networks that is truncated through a combination of finite size and a scaling condition. Our work extends the results of Ginzburg and Sompolinsky (1994) by providing the systematic higher-order expansion without explicitly requiring the consideration of the rest of the hierarchy. We also provide conditions for the truncation of the expansion and consider the network response to correlated input. Our expansion is not a finite size expansion, although it can reduce to a finite size expansion under certain conditions (such as normalized all-to-all connectivity in the network).
In section 2, we revisit the original Wilson-Cowan framework and propose a Markov model that has the minimal stochastic dynamics to produce the Wilson-Cowan equations. This will be more rigorously justified in section 4. Section 3 presents the derivation of a moment hierarchy for this Markov model. After truncating, we provide a posteriori justification for the truncation. It will be seen in section 4 that the validity of this truncation was in fact natural and did not require ad hoc assumptions. The truncation conditions turn out to be related to the proximity to a bifurcation point as well as the extent of connectivity in the network. We also make more precise the sense in which our Markov model is “minimal” by introducing the path integral formulation. The field theory formalism that appears in this letter arose in the context of reaction-diffusion problems. (See Janssen & Tauber (2005) and Tauber, Howard, & Vollmayr-Lee (2005) for reviews of this formalism applied to reaction-diffusion and percolation processes.) We demonstrate a simple example all-to-all system in section 5 and show some simulation results.
2. Rate Equations Reconsidered
We imagine that the typical rate equation is produced by some marginalization process over both disorder and extra degrees of freedom. Hence, it may be possible to derive a generating function for the statistics of the marginalized process. The lowest order in the steepest-descent expansion of the generating function describes mean field theory, which gives the rate equation. Since the operation of marginalization is dissipative, we cannot recreate the underlying microscopic process exactly with only the rate equation alone. However, the mean field theory places constraints on the structure of the dynamics, enabling us to investigate the structure of higher-order statistics implied by the structure of mean field theory. In the original derivation by Wilson and Cowan (1972, 1973), the activity variable was presumed to describe the fraction of neurons firing per unit time within some region of the brain. Two main features of this interpretation bear emphasizing. First, the rate equations were originally understood to be equations providing the dynamics of the probability that a neuron at x will fire at time t. There is therefore an implied underlying probabilistic model. Second, the probability a(x,t) applies to all neurons within some region of the brain, not just a single neuron. Thus, a spatial averaging component is implicit in the equations. The original Wilson-Cowan rate equations thus described the dynamics of the probability for a neural aggregate in the brain. Another feature implicit in the Wilson-Cowan equations is that these probabilities are independent for each neuron. This implies that the Wilson-Cowan picture is one in which neurons fire with Poisson statistics with firing rate determined by a(x,t), a picture supported by neural recordings (Softky & Koch, 1993).
Given this perspective, one might consider what processes may underlie rate equations. One route is to treat the fundamental, small-scale dynamics as a probabilistic process, for example, a Markov process. In this case, the basic description for neural activity will be provided by a master equation governing the evolution of probabilities for different neural configurations. This route obscures the source of uncertainty in neural activity in favor of directly modeling the probabilistic activity. This tactic has been used to model the so-called asynchronous irregular states seen in some neural models (Van Vreeswijk & Sompolinsky, 1996; El Boustani & Destexhe, 2009). Another route would be to employ the strategy of kinetic theory (Nicholson, 1992; Ichimaru, 1973) and define a continuity (i.e., Klimontovich) equation for the probability density of a network of deterministic neurons. (For an example of this approach applied to coupled oscillators, see Hildebrand, Buice, & Chow, 2006; Buice & Chow, 2007.) In that work, the probabilistic aspects of the model arise from the distribution of driving frequencies and initial conditions. Ultimately the difference in the two approaches is the origin of stochasticity—whether it is implicit in the dynamics of the neurons or an emergent property of the interaction of deterministic neurons (e.g., chaos). In either case, the final product is an effective stochastic dynamical system. In this letter, we will follow the approach of assuming an underlying probabilistic model given by a master equation, so that any emergent chaos has already been absorbed into the dynamics. We then seek a minimal stochastic model that will produce the Wilson-Cowan rate equation at the mean field level. We can then formulate equations governing the fluctuations of this model. In this section, we motivate such a minimal model qualitatively, leaving a more rigorous approach for section 4.
is a measure of the mean activity in the network. We obtain an equation for ai(t) by multiplying equation 2.2 by ni and taking the sum over all configurations . However, this equation is not closed (i.e., it depends on the second and possibly higher moments). An equation for the second moment can be similarly constructed by multiplying equation 2.2 by ninj and summing over all configurations. The resulting equation will depend on the third and higher moments. Continuing this process will result in a moment hierarchy with as many equations as there are locations, which could be infinite. In general, no finite subset of this hierarchy is closed. This means that if we wish to have a closed set of equations, we need to make some approximation that allows us to truncate the hierarchy.
The moment hierarchy approach does not make any approximation. It is a change of variables from the distribution to moments of that distribution. The approximation arises when we truncate this hierarchy in order to render the equations tractable. The simplest truncation is mean field theory. The first-order corrections to mean field theory are given by truncating at the next order. Truncation of the moment hierarchy requires some justification. We will demonstrate below that this justification in the neural case may be provided by the large spatial extent of neural connectivity and the distance of the system from a bifurcation.
3. Truncation of the Moment Hierarchy
An immediate noteworthy consequence is that Cij(t) will have substantial input only when the activity is such that f′(s) is large. As an example, suppose f(s) is a simple sigmoid function. In this case, f′(s) is peaked at threshold (where we define threshold to be the central point of half maximum) and zero far away from threshold. Reasonably we have the result that correlated activity will increase only when the input to a neuron is near threshold. If the slope of the sigmoid is such that f(s) is a step function or near a step function, then Cij(t) will receive input only when the activity is precisely near threshold. Also notice that the strength of the input to Cij(t) is proportional to the weight wij between the neurons in question, as well as the mean activity. An initial check on the equations is that Cij decouples from ai in the case where f(s) is linear or constant.
3.1. Criticality and Truncation of the Hierarchy.
Although we have derived equations for the mean activity and equal-time correlation, there are some outstanding issues. The primary concern that must be addressed is that we require some justification for the truncation of the hierarchy at the level of the two-point correlation function Cij(t) instead of allowing higher moments to interact with the mean activity.
We have two competing effects. On one hand, we have the system size governing the magnitude of correlations. On the other, the distance of the system to a bifurcation likewise affects the size of fluctuations. The relative trade-off of the two, from the definition of C0ij, is given by the product of the smallest eigenvalue of Γij and the number of presynaptic neurons Ni. We will demonstrate this relationship more precisely when considering the all-to-all network in section 5.
Finally, it is worth pointing out the effect of input upon the hierarchy. If the input is another Poisson process, then the only equation affected is the one for ai(t). The higher equations in the hierarchy are affected by this input only through its effect on the mean activity ai(t) and the firing rate function f(s). In general, this suggests that large external inputs will actually reduce fluctuations, depending on the form of f(s), in the sense of driving the system toward Poisson-like behavior, a reasonable result. In particular, if f(s) is a saturating function, then the correlations will decouple from the equation for ai(t) and the source terms for higher correlations will be driven to zero, leaving the system described completely by the rate equations. The analogous situation for a ferromagnet is driving the system with a large external magnetic field.
4. Path Integral Solution of the Master Equation
We have thus far demonstrated how a minimal Markov model consistent with the Wilson-Cowan rate equation can be used to derive generalized equations in a hierarchy of moments. Although we truncated this hierarchy at second order, one can in principle truncate at any desired cumulant, although the calculations become successively more cumbersome. Here we show that the moment hierarchy is equivalent to a path integral or field-theoretic approach, which systematizes the perturbation theory for the statistics of the network by providing rules for the construction and evaluation of the cumulants. Another major benefit is that it provides a systematic means for obtaining moment truncations or closures. The path integral representation of the master equation, 2.2, was derived by Buice and Cowan (2007) by modifying a method originally developed for reaction diffusion systems (Doi, 1976a, 1976b; Peliti, 1985). We quickly review the representation and then detail how the generalized equations can be derived from this representation.
4.1. Closed-Activity Equations from the Path Integral.
The importance of this approach is that as we consider successive orders in the loop expansion, the effective action closes the system automatically. If we could calculate Γ[aμ] for our model of interest, then we would have the exact equation of motion for the true mean of the theory. In essence, we are trading a closure problem for an approximation problem. The advantage gained is that we do not have to worry about the contributions of higher moments explicitly and can consider explicitly the criteria allowing us to truncate the expansion. If there is an explicit loop expansion parameter, this truncation is straightforward. If not, as in our case, we must explicitly assess the regimes in which any truncation is valid. Even in the case where a truncation fails, the loop expansion can provide guidance in terms of identifying classes of diagrams (i.e., terms in the expansion) that are relevant in appropriate limits, which could be summed.
With the moment hierarchy approach, in order to produce better approximations, we are required to truncate further in the hierarchy. This can quickly produce unwieldy equations. The loop expansion provides an alternative in that corrections to the generalized equations can be produced with a diagrammatic expansion, namely, the one that calculates Γ2[aμ, Cμν], from which the corrections to the equations can be calculated.
5. All-to-All Networks, Finite Size Effects, and Simulations
Importantly, we see that we can alter the bifurcation structure by adding a forcing or source term to the correlation function C(t) equation, linearly shifting the C nullcline. This removes the stable fixed point for high a (the one associated with the activated state in mean field theory). Because of this, we see that we can disrupt the activated state by stimulating the system with an input such that the correlation is sourced more strongly than the mean. We can use this to “turn off” the activated state by synchronizing the network. These correlations drive the system to the absorbing state of the full model. To reverse this deactivation, we simply drive the system with Poisson noise (i.e., force the equation for a(t) but not C(t)). Compare this to the effect demonstrated in Laing and Chow (2001) in which synchronized activity associated with fast synapses led to the destabilization of activity that the Wilson-Cowan equation predicts to be stable. For a saturating firing rate function (more generally, a function such that f′′(s) < 0 in the appropriate region), increased correlations inhibit the mean activity ai(t).
The plots for N = 10 in Figures 6 and 7 demonstrate the breakdown of the generalized equations. There is already a significant deviation of both mean field and the generalized equations at α = 0.5. Naturally the discrepancy is accounted for by the poor estimation of the correlation at this level. As we near α = 1.0, the estimate of the correlations begins to grow, whereas the simulated correlation is dropping to zero.
We have demonstrated a formalism for constructing a minimal extension of a rate equation to include correlated activity. We have explicitly computed the lowest-order results of this extension, which provide coupled equations for the mean activity, two-point correlations, and linear responses. These results indicate that correlations can have an important impact on the dynamics of a rate equation, affecting both stability and the structure of bifurcations. Our argument relied on inferring a “minimal” Markov process. Our use of the Doi-Peliti path integral formalism guides our assertion that our inferred Markov process is the simplest one compatible with both the rate equations and their interpretation as measuring some stochastic counting process. Thus, a general extension for any type of rate equation should share the same basic structure that we have described here. We performed this construction on a Markov process consistent with the Wilson-Cowan equation, but our prescription would work equally well with any Markov process.
In keeping with this idea, our results have something in common with other approaches to understanding correlations in neural networks. El Boustani and Destexhe (2009) attempt to derive a Markov model for the asynchronous irregular states of an underlying neural system and explore the moment hierarchy of that Markov model. We take the opposite approach, beginning with a presumed set of rate equations and asking what possible restrictions can be placed on the correlation functions knowing only the dynamics of the rate model. Their hierarchy is truncated via scaling and finite size, whereas our hierarchy's truncation (and the truncation of the loop expansion) arises through the distance to a bifurcation in the rate equations. Ginzburg and Sompolinsky (1994) propose a simple Markov model and study its moment hierarchy. For the correlations, they achieve results similar to the tree level of our loop expansion. An important point of departure is that we consider the recurrent effects the correlations have on the mean field, which we demonstrate can be sufficiently significant to alter the structure of the bifurcation.
As we predict, our theory breaks down sufficiently close to a bifurcation. Examining the dynamics near the critical point requires a different form of analysis such as a renormalization argument. An example was presented in Buice and Cowan (2007), where it was argued that a large class of neural models would exhibit scaling laws near a bifurcation like those of the directed percolation model (Janssen & Tauber, 2005). The predictions of this scaling coincide with measurements made in cortical slices of “avalanches” (Beggs & Plenz, 2003). If criticality is important for neural function (Beggs, 2008), then these sorts of approaches will be more important for future work, and our rate model extension will be less applicable.
In contrast, supporting the potential usefulness of our rate model extension is the fact that large neural connectivity suppresses correlations and aids the truncation of the hierarchy, an analogous result to the Ginsburg criterion in equilibrium statistical mechanics. In addition, we demonstrated that Poisson-like input in general pushes the system into configurations in which the correlations are suppressed relative to the mean. All of this suggests that our extension will be at least as applicable as the rate models themselves.
Regarding that applicability, both the Markov process and the rate equations assume a large degree of underlying asynchrony in the network. The expansion we describe should be appropriate for networks in which a relatively small amount of synchrony at the level of individual neurons is developing. The coupled correlation function captures this effect. If the population is being dominated by neuron-level synchrony, then the Markov process should no longer hold as a description of the system. Population-level synchrony as captured by the original rate model, however, should have no effect on our analysis. In other words, there will be correlation-effects acting on oscillating states, for example, such as presumably correlation-induced modulation of the frequency of the oscillation. We will demonstrate this in future work.
An important outstanding point is that we have posited this Markov process based on the original interpretation of the Wilson-Cowan equations as dynamical equations for the probabilistic activity of neurons. Although our Markov process is the most “natural” given the transitions in the Wilson-Cowan equations, there is no a priori reason to suppose that this Markov process reflects the probabilistic dynamics of a physiologically based neural model or of real neurons precisely because there is nothing that mandates this interpretation. Per the renormalization analysis of Buice and Cowan (2007), measuring scaling laws in cortex will provide a means of identifying classes of models (by identifying the relevant universality class), but this will in no way distinguish among models within the same class. Distinguishing models within the same class will require the measurement of nonuniversal quantities. This would likely mean some relatively precision measurements of response functions in cortical activity.
Our model is one in which activity is Poisson distributed in the absence of connectivity. Likewise, we have shown that connectivity, away from a bifurcation, only slightly modifies this. Although we introduced the model in terms of populations, its original description arose in the context of the low firing rate reduction of the Cowan two and three state neural Markov processes (Cowan, 1991), where ai(t) is the probability of neuron i being active at time t. The unboundedness of the variable ai(t) arises from the low firing rate assumption. In Buice and Cowan (2007), the model is described as counting the number of recent spikes from a single neuron at i that still produce an effect upon the activity of the network; for a given spike, this effect decays with rate constant α. Either of these descriptions provides justification for the Poisson nature of solutions in terms of single neuron activity. If we have large numbers of equivalent neurons, then an averaging process reduces the effect of correlations, consistent with what we have shown here. Our all to all example produces a network in which the deviations from Poisson are governed by finite size effects. A population average over such equivalent neurons will reduce the fluctuations according to the size of the population. In the limit of an infinite number of identical neurons, we are left with mean field theory.
We feel our approach is a useful starting point for understanding effects beyond mean field. We have demonstrated a correlation-induced loss of stability in an all-to-all network. This effect should carry over to nonhomogeneous solutions such as bump solutions or traveling waves. Likewise, correlations will modify important aspects of mean field solutions such as dispersion relations. Our approach enables this dispersion relation to be calculated. In addition to stability, our equations are a useful starting point toward understanding the wandering of bump solutions. They also provide a natural means of studying beyond mean field effects of correlation-based learning. A model of visual hallucinations in cortex based on the Turing mechanism has explained many hallucinatory effects (such as the various Kluver form constants). Since the Turing mechanism is based on bifurcations, it is an interesting question to what extent the coupling with correlations affects the results of the hallucination analysis. Our approach may provide this model with a means of explaining further hallucinations not covered by the model in Bressloff et al. (2001).
It remains an important question how to connect our Markov and generalized rate model approach with models of deterministic neurons. While the formalism admits almost any gain function, there remains the question of connecting this gain function to, for example, the transfer function for some neural model of which the Markov process is some approximation. This is, of course, not a question of the analysis of Markov models but of the applicability of rate models as high-level descriptions of more detailed neural models. Answering this question will likely involve a kinetic theory formulation of the neural systems, such as the one pursued in Hildebrand et al. (2006) and Buice and Chow (2007). Having derived the generalized equations, it is also now important to explore their further consequences for phenomena such as pattern formation. There are also many avenues to extend this model and this approach. The Markov process can be enlarged to account for synaptic adaptation by adding a synaptic time variable to the neural configuration. Likewise, noise in the transitions themselves, whether spatial or temporal, is easily incorporated into the action. A reduction of the resulting theory would no longer satisfy the Markov property, although there may be certain natural assumption (such as slow dynamics for the auxiliary field) that could allow one to regain Markovicity with an approximate model. This would allow us to construct extended Wilson-Cowan equations that incorporate these and other aspects of neural dynamics. These questions will be explored in future work.
Appendix A: Composite Operator Effective Action and the 2PI Equations
We can use the loop expansion to draw some conclusions about the applicability of perturbation theory. Since Γ2[aμ, Cμν] is second order and is the sum of vacuum two-particle irreducible graphs, every graph contributing to it must be at least of two loop order. Every internal line represents a factor of Cμν, and so each graph contributing to Γ2[aμ, Cμν] must have at least two factors of Cμν, each of which will be either equal to 0 or be attenuated (in steady state) by the same exponents that attenuate the magnitude of C(x, y, t) away from a bifurcation, according to equations B.14 to 4.36. Thus, the argument that Cμν is small away from the critical point extends to every term in the expansion for the generalized equations.
The caveat here is that there is a class of diagrams that couple the lowest-order expression for a given moment to the mean field. Although these graphs are suppressed by the distance to criticality, each of these is of the same order. We are assisted by two facts. The first is that the source terms for each of these moments at lowest order will be proportional to derivatives of the firing rate function. If f(s) is sufficiently smooth, this will suppress higher-order contributions. In addition, each coupling will go as an additional factor of N−1m where Nm was defined in section 3.1 as the smallest number of inputs to any given neuron. Thus, the connectivity in cortex will serve to “average-out” sources to the mean from higher moments. This will be the case as long as we can bound the total input to any given neuron.
Appendix B: Tree Level Equations of Motion
In order to calculate the expansion for the equations of motion, we need to compute the value of both Lμν and Γ2. We compute the lowest-order correction here.
The “mean field” portion of the equations of motion, A.16, are obtained from equations B.1 and B.2 (by setting the left-hand side to zero). The remainder of the equations of motion are “classical” terms dependent on the correlation functions and loop corrections. The latter are given by the term in equation A.16. The term in the trace is, of course, the sum of the left-hand side of equations B.7 and B.8.
We can simplify the equations for the mean field by realizing that any term involving C−1,1 or C1,−1 can be ignored because they will appear only in the form C−1,1(x′, t; x, t), that is, at equal initial and final times. These will be zero. This can be seen as either the “initial condition” for the linear response terms or as a manifestation of the “ϵ(0)” problem in quantum field theory (Zinn-Justin, 2002).
We thank Vipul Periwal for helpful suggestions. This work was supported by the Intramural Research Program of the NIH/NIDDK.
“2PI” stands for 2 particle irreducible. The effective action Γ[aμ] is the generating functional of 1PI graphs, that means that the graphs that determine Γ[aμ] cannot be disconnected by cutting any single line of the graph. Similarly, Γ[aμ, Cμν] is 2PI in the sense that graphs contributing to it cannot be disconnected by cutting two lines.