Detailed network simulations
We developed a simulation program based on the thalamocortical column model described in Traub et al. (2005), with this major difference: The only cell type now simulated was the layer 5 tufted intrinsically bursting (IB) pyramidal cell, of which there were now 15,000. Each model neuron had, as before, 61 compartments: a soma; a 6-compartment axon; and 54 compartments for the basal, oblique, and apical dendrites. There were multiple voltage-dependent and Ca2+-dependent membrane conductances, although for the sake of simplicity, high-threshold gCa and gK(AHP) were blocked.
The model neurons were interconnected by axonal gap junctions, in a globally random topology, with an average of 3.33 gap junctions on each neuron (written as <i> = 3.33, the symbol standing for “mean index”). Gap junction conductance was 7 nS in the example illustrated below, which would usually permit an action potential to cross from one firing axon to a coupled other axon. Random (i.e., following independent Poisson statistics) current pulses were delivered to each axon at an average frequency of 4 Hz; such pulses would cause “ectopic spikes” when the cell was not refractory from a recent spike. There were no chemical synapses.
The program stored membrane potentials at various locations (axon, soma, apical dendrite) of selected neurons, as well the estimated field potential, the units of which are arbitrary. The method for estimating field potentials is described in Traub et al. (2005).
Each run of the program simulated 1 s of neural activity. The program ran on 40 processors of an IBM (Armonk, NY, U.S.A.) Linux cluster, and required approximately 36 h to complete. Code was written in Fortran and run in the message passing interface (mpi) parallel environment. Cellular automaton code (see below) was also written in Fortran. Smaller models ran on a single processor of an IBM 7040-681 AIX parallel machine, whereas the largest model (5.76 million cells) ran on 20 processors of the same computer, again in the mpi parallel environment. All source code is available upon writing to email@example.com.
Cellular automaton model
Cellular automata (“CA”) (Sarkar, 2000; Wolfram, 2002) have been used to model a number of phenomena and processes in mathematics and physics, including the behavior of gases, galactic structure (Seiden et al., 1979), and neuronal networks (Pytte et al., 1991). First, we describe the simplest form of CA—finite deterministic—and then describe the stochastic modification actually used in the report.
A finite, deterministic (synchronous) CA consists of a finite number of elements, here called “cells” to suggest the biologic connection, where (1) each cell has inputs from a specific set of other cells; (2) each cell can exist in a finite number of states, the list of states being the same for all cells; (3) time is discrete, and cells change states synchronously every time step; and (4) there is a rule—the same for all the cells—that determines how cells change states: the rule has the property that state changes, for each cell, depend only on the present state of the respective cell, and the present states of cells supplying input to the respective cell.
With the stochastic modification that we use, the state-change rule also allows cells that are in a particular state, to switch states randomly, according to a defined statistical rule. This stochastic property corresponds biologically to the possibility of a spontaneous action potential occurring in an axon, perhaps as a result of fluctuations in Na+ channel openings.
The “inputs” in the CA model correspond to connections mediated by nonrectifying gap junctions, and are, therefore, assumed symmetrical: If cell A receives input from cell B, then B receives input from A. The states will correspond to “firing,” various stages of “refractoriness,” and “excitable”—that is, available to be excited if a connected cell is firing, or if a spontaneous event occurs. The discrete time step in the CA model corresponds physically to the shortest relevant time interval for VFOs: the time it takes for a spike in one axon to evoke a spike in a coupled axon (this time, in the actual brain, will depend on gap junction conductance, intrinsic membrane properties, and other quantities). We shall assume that the discrete time step in the CA model corresponds, physically, to 0.25 ms, which is biologically reasonable (Mercer et al., 2006). The CA rule for changing the states of cells will incorporate the idea, from membrane biophysics, that after a cell fires, it is refractory for a short time, and then available to fire again. (The CA model used here is too simple to incorporate those details of membrane kinetics that actually govern refractoriness.)
More formally, the CA model is built, and behaves, as follows:
(1) Cells are usually organized into a two-dimensional (“2D”) array, either 400 × 300 (120,00 cells), or 800 × 600 (480,000 cells). The unit spacing in the array will be called the “lattice spacing”; what physical dimensions to which the lattice spacing corresponds will be considered in the Discussion. The shape of the array, with one side = ¾ × the other side, was chosen to correspond to the 6 × 8 subdural grid of electrodes used in patients. In some simulations, we used a larger 3D array, with dimensions 1,600 × 1,200 × 3, containing a total of 5,760,000 cells. Note that the clinical ECoG data available to us are effectively 2D, motivating our preferred use of 2D model arrays; 3D arrays were tested for the sake of completeness.
(2) The statistical properties of the connectivity are defined by two parameters: (a) the total number of connections (i.e., “gap junctions”), or alternatively <i>, the mean index; for the simulations illustrated in this article, <i> = 1.33, a sparse degree of connectivity; and (b) a parameter cr, called by Lewis and Rinzel (2000) the “footprint.” Two cells are allowed to be connected to each other if, and only if, they are at most cr lattice spacings apart. (Of course, two nearby cells will, in general, not be connected: Only some allowable pairs actually have connections.) When cr = ∞, the graph is “globally random.” In this case, the only parameter determining connectivity is the mean index, <i> (c.f. Erdös & Rényi, 1960, who use a different notation, namely what they call “c,” equal to 0.5 × <i>). cr is a critical parameter for our model, determining whether waves of activity exist, and how fast they will propagate. For the 3D model, cr depends only on separation in the x–y plane (1,600 × 1,200), and not along the z axis.
(3) Each cell can exist in any of 17 possible states, which we may call “firing” (or “on”); “excitable”; and 15 refractory states, called refr1, refr2,…, refr15. (Thus, the total refractoriness will be 15× the time step = 15 × 0.25 ms =3.75 ms.) The refractoriness was made this long to prevent epochs of re-entrant rhythms caused by activity propagating around short cycles in the network, leading to extremely high-frequency stereotypical activity (Lewis & Rinzel, 2000).
(4) The dynamics, that is, the rules for changing states, can be divided into two parts. The first part is, in a sense, trivial: It consists of state changes that are “forced,” that is, that do not depend on inputs to the respective cell. These state changes are: firing refr1; refri refri+1 (i = 1, 2,…14); refr15 excitable. These state change rules say, in effect, that a firing cell goes through a determined period of refractoriness, during which it cannot be forced to fire again, either by inputs from other cells or by a spontaneous event. The second part concerns the possibilities for an excitable cell: (a) if any cell connected to an excitable cell is firing at time step t, then the excitable cell firing at time step t + 1; (b) if a spontaneous event occurs (see below), then the excitable cell fires at the next time step; (c) otherwise, the cell simply remains excitable. With respect to condition (a), what this means is that the probability of propagation of spike from one axon to another is unity, provided that the second axon is not already firing and is not refractory. In preliminary simulations (not shown), we checked that if the propagation probability is less than unity (but not too small), then network frequency is not systematically altered; but we did not pursue this question as regards spatial characteristics of the oscillations.
The spontaneous events are characterized by a single parameter, pspon, the probability that an excitable cell fires spontaneously on a given time step. A typical value of pspon would be 1/80,000: This value would imply that an excitable cell fires—on its own—an average of once every 20 s.
To summarize the basic parameters: If the array dimensions are fixed (which also determines the total number of cells), and the rules for state transitions are fixed, then the three basic parameters are: <i> (the mean index, which may also be specified by stating the total number of gap junctions in the network); cr, which determines how localized the connections are in space; and pspon.
Simulations were run in either of two modes: first, pspon was set to 0, and a single cell set to firing, while all other cells were excitable. In this way, one could observe the behavior of a single wave of activity, unperturbed by other waves. Alternatively, all cells were initialized as excitable, with pspon at some fixed nonzero value. In this way, spontaneous activity was simulated, for up to 8,192 time steps. The following quantities were stored: the total number of firing cells, in the whole array, and in each of 48 equal-sized square subarrays, in a 6 × 8 arrangement (to match the subdural grid layout used for the illustrated patient data). For making videos of the activity, the locations of firing cells were stored, every 2–5 time steps; however, to make the images intelligible, only every fourth or every sixth firing cell was saved. When only a single wave was studied, the computer saved the following quantities as functions of time: the total number of firing cells, the mean distance of all firing cells from the initially firing cell, and the standard deviation of this latter quantity. Long runs of spontaneous activity were also analyzed with a fast Fourier transform algorithm.