2.3. Part 3: Simulation of Adult ERP Data

In what follows, the descriptions of the components of the analysis pipeline for the third part of the study are organized according to the sequence of steps illustrated in Figure 1. From the information derived from the ICA (Step 1) and the estimated dipoles (Step 3), we derived simple single spiking representations adopting electric-fields estimation modeling (Step 5). Next, the results of Step 5 were used to implement an ERP-based Adaptive Control of Thought—Rational (ACT–R) modeling approach previously validated on the same task [43,44]. We organized the simulated dipoles and the simulated ERP spikes in patterns or chunks of activity corresponding to cortical areas postulated in the adult ACT–R (Step 6). The results of Step 6 allowed us to simulate the sequence of spatiotemporal activity as dipoles mapped onto the same adult structural MRI template as the children’s data (Step 7). This was the basis for comparing children’s and simulated adult estimated localization, therefore, testing for structural homology.

The results of Steps 6 and 7 were also used for aggregate series of simulated spikes to build polyspiking patterns for the simulated dipoles (Step 8). Finally, in Step 9, the simulated data obtained from Step 8 were converted to ERP topographic maps. This final step made possible to contrast children’s and simulated ERP topographic mappings, therefore, testing for functional homology.

To simulate the electrical activity, each module in the neurocognitive model was assumed to be generating one or two dipoles in the location identified in the dipole-fitting stage. The module was assumed to produce its electrical energy in a rising and falling spike. For modeling purposes, a simple triangular wave was assumed, which peaked at the center of the module. The resulting electric field (voltage) was then calculated at the surface of the head for each electrode as the sum of the individual dipole contributions. Elsewhere, we have shown that this method generates reliable, valid and consistent descriptions of actual ERP activity [43,44]. Since the spiking activities of the components occurred at different times in the observed data, it was not necessary to add the effects of more than one dipole at a given time.

The effect of each dipole was estimated in the simulation by following three steps (see Figure 3): (i) The square of the distance r from the dipole to an electrode was calculated by using Pythagoras. Next, (ii) the cosine of the angle θ between the electrode and the dipole was calculated by using vector dot product. Successively, (iii) the electric potential from the dipole at the electrode was derived from Coulomb’s law (=k.p.cos(θ)/r²), where p is the strength of the dipole and k is a constant. (It was not necessary to know the value of the constant since relative magnitudes were used in the model).

Schematic representation of the calculation of an electric dipole field spiking by simulation; the left panel shows the output of the calculated electric-field potential represented as a simplified spike at fixed timing (determined by the simulated production schedule, here shown at an arbitrary time point for example sake; for the actual simulated timings in the present study see the production schedule shown inTable 1).

To model and simulate our data, we used an adapted version of John R. Anderson’s Adaptive (Control of Thought—Rational) ACT–R [64]. In the general architecture, cognition is considered to arise from the parallel interaction of several independent modules. However, top-down processes are directed by the Procedural Module, which is meant to model procedural memory. ACT–R models procedural memory as a production system. Specifically, procedural memory contains production rules (i.e., if/then rules). Communication to and from the Procedural Module is managed by buffers and chunks (see Figure 4). Chunks in ACT–R are lists of predicated information (for example, “duck” could be represented by the chunk: Is-an:animal, Name:duckling, Color:yellow, Size:small).

The organization of information in ACT–R (Adaptive Control of Thought–Rational). Adapted with permission from [42].

Each buffer can contain one chunk at a time. Each module has at least one buffer, so there is a visual buffer, a declarative memory buffer, and so on. Modules receive instructions from their buffers and place the results of their activity in their buffers. Collectively, the buffers can be thought of as working memory; they can also be thought of as representing the current context of the task. Productions “fire” when their “if condition” matches the contents of the buffers. The “then” part of a production then alters the content of the buffers. Productions can only fire one at a time.

In our version of ACT–R, the productions represent electric-field potentials. We programmed the simulation to fit the midpoint of the vincentized bins used to parametrize the ERP data series, so that each production was conditioned to occur at successive steps of approximately t_i + 50 ms, with i = 0, 100…, 600. The 50-millisecond cognitive cycle is assumed in many realistic modeling architectures besides ACT–R (for example, Soar, EPIC, GOMS, see [65] for discussion). The neurobiological plausibility of this 50-millisecond cycle has been demonstrated by spiking neural network models simulating well known time constants for the GABA-A receptors in the Basal Ganglia, which in ACT–R (and in various other architectures) is assumed to be responsible for the working memory central executive [66]. Therefore, in the practical implementation of the model we assumed a base production-completion time constant which corresponded to the size of the vincentized bins (100 ms) plus the 50-millisecond cycle constant. Each module contained functions (specifically, ex-gaussian convolutions) to determine activation levels during rest, during the task and the decay rate after firing. Effectively, these functions determined the spiking behavior associated with a given production: Faster productions corresponded to more intense and more quickly decaying spiking.

To implement the simulation, we adopted Python ACT–R [67]. In particular, the present version of the model assumed that the caudate in the basal ganglia acts as the central coordinator (executive) of productions. The hippocampus controls declarative memory while the cingulate cortex controls attention to conflicting stimuli. Frontal cortex supports declarative memory while visual processing takes place in the occipital with further processing in the parietal (see Figure 3). With the time constraints as shown in Table 1, we implemented an ACT–R model which predicted that initially the visual module (occipital) would be activated by the displayed pictures of the target (duck) and turtle (distractor) and would place a representation of the picture in the visual buffer (parietal). Next, the “parietal” representation would be used to retrieve the label of the object and the appropriate instruction about what to do in response to the image of that animal from declarative memory (temporal), which in turn would be placed in the planning buffer (frontal) and initiate the motor program for the manual response, or stop it (basal ganglia).

Simulated locations and predicted timing of spiking occurrence according to the present adapted Python ACT–R architecture.

Note: The timings shown in the table include the base constant and the increments from the 50-millisecond cognitive cycle (described in Section 2.3.2) and each of them represent the approximate estimated moment in which a given production process in the ACT–R model is completed.

As in the case of the actual children’s dipoles, the ACT–R simulated dipole data was mapped on a standardized MRI template provided by the Yale Bioimage Suite Web [63].

In this step, the simulated electric-field spikes were chunked in ordered series corresponding respectively to the standard localizations and predicted timing of spiking assumed in the programmed schedule for the firing of ACT–R productions (see Table 1). The output of this step was a set of polyspiking patterns including a minimum of six spikes for each of the twelve electrodes. Within each pattern, the spike with maximal activation derived from the dipole of interest at a given bin interval, while all other spikes reflected resting or decaying background activations from the other dipoles. The latter configuration permitted to obtain coarse coding which could quantify the amplitude of each spike in the aggregated pattern as a color category (see example in Step 8, Figure 1) corresponding to a standard RGB value so that it could be ordered along an intensity scale.

The simulated topographical maps were obtained through the same utility of EEGLab as the one used for the actual ERP data. To feed the simulated data in a compatible format, the values of the color intensities corresponding to the polyspiking patterns were previously converted (in Stage 8) into an arbitrary relative voltage scale (with range from blue/black, −6 µV, to red/orange, 6 µV).

To run the structural comparison, we first estimated the margin of localization error by computing the range differences based on the matches resulting from the search nearest grey area procedure; the measures were in millimeters. Successively, we computed z-scores of these ranges (henceforth called z-Ranges) which permitted to compare the extent of variations in localization in the actual children’s data against those in the adult simulated estimations.

ERP activity comparison between the actual children’s topographic maps and the topographic maps derived from the simulated ERP activity were computed as linear regressions of vincentized averaged ERP amplitudes in the topographic map of children against the corresponding ACT–R simulated data; the fit was assessed by calculating the coefficient of determination, R².