Field-theoretic approach to fluctuation effects in neural networks

A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification of dynamic universality classes in vivo is an outstanding and important question for neuroscience.

Figure 1

Graphical representations of Cowan dynamics for the (a) two- and (b) three-state neural models. $A$, $Q$, and $R$ represent active, quiescent, and refractory states, respectively. The arrows represent possible transitions with the indicated transition rates. Open arrows indicate that the corresponding rate is potentially dependent on the state of the network, i.e., the activity of the other neurons in the network, with firing rate functions $f$ and $g$.

Figure 2

Feynman graph for the propagator. Time increases from left to right so that $t^{\prime }<t$.

Figure 3

Feynman graph for the moment $⟨\phi (x_1,t_1)\phi (x_2,t_2)\tilde{\phi }(x_3,t_3)⟩$.

Figure 4

Feynman graph for the mean with no interactions and a simple linear initial condition, represented by the bold dot.

Figure 5

First four terms in the perturbation expansion of $⟨\phi (x,t)⟩$ using only the vertex ${\mathcal{V}}_{02}$. The dots represent the initial condition operator. Time moves from right to left.

Figure 6

Equation of motion for $a(x,t)$, exact to all orders. Shown are the linear, quadratic, and cubic parts of the equation. The gray circles represent the proper vertices which are given by the loop expansion of the effective action. Solid lines indicate $a(x,t)$ to all orders; dotted lines indicate the propagator from Eq. (47).

Figure 7

One-loop approximation to the quadratic term in the equation for $a(x,t)$. There are similar diagrams for the cubic and higher terms as well, all of which sum (along with the linear term) to give the term $\mathcal{N}$ in Eq. (49).

Figure 8

Lowest-order contributions to ${\Gamma }^{(1,1)}[0,0]$. When the one-loop contribution is of the same magnitude as the mean field propagator, mean field theory is no longer dependable.

Figure 9

Tree-level graph for the two-point correlation function $C(x_1,t_1;x_2,t_2)$. It should be compared with the diagram of the loop correction to $a(x,t)$ in Fig. 7.