Figure 1.
Multiscale dynamics: slow operational signal and SFM emergence.
Panel A: The slow operational signals {ξj} converge through a fast transient to a specific ξj node resulting (here) in the emergence of a cylindrical manifold. Panel B: The functional dynamics {ui} collapses fast (also) onto the manifold where it executes a slow spiral flow. The ξj node's stability is sustained for the duration of the flow execution. Subsequently, the ξj node destabilizes, followed by the related manifold and the dynamics moves away, again through fast transients. The density of data point is inversely proportional to the time scale of the dynamics.
Figure 2.
Functional modes and architecture overview.
Interactions among functional modes Fj({ui}) result in one of them dominating the output of the functional architecture for a period of time followed by the domination of another one via a fast transient. Three modes (associated with different colors) of the available dynamic repertoire are shown. They all correspond to 2-dimensional phase flows on an ellipsoid manifold. Blue represents a monostable phase flow, magenta a limit cycle flow, and green a bistable flow. Their vector field and a set of characteristic trajectories starting from different initial conditions (small asterisks) are shown. The modes' mutual interactions are depicted as bidirectional couplings (arrows) among their associated weighting coefficients {ξj(t)} (with which they have a multiplicative relationship). The resulting expressed phase flow F({ui},t) (shown as a trajectory in the phase space and time) results from their linear combination at each time moment, while {ui(t)} is the respective time series. {ξj(t)} play the role of a slow operating signal (with respect to the inherent time scale of the functional modes, i.e., τξ>>τf). Finally, an instantaneous (τδ<<τf) operational signal {δi(t)} (in red) may have an additive contribution to F({ui},t), acting like a meaningful perturbation.
Table 1.
Variables, parameters and time scale hierarchy.
Figure 3.
Interactions among the functional architecture's components.
The functional dynamics F({ui},{ξj},{δi}) is fed back to the slow serial dynamics by means of inhibitory feedback, Finh({ui},ξj,{λj}), and is integrated by the feedback integrating variable vj. vj causes timely fast transitions to the ‘switching’ variable λj to which it is coupled via the sigmoidal function S(vj) and excitatory feedback Fexc(λK’). λs regulate the WTA competition of the {ξj} (via Lj({λj}) and Cj({λj})) that determine which functional mode will dominate the expressed phase flow F({ui},{ξj},{δi}) at each time moment. The instantaneous operational signal {δi({ui},{ξj})} also receives feedback from output {ui} and is also coupled to the {ξj}. Thus, the whole functional architecture becomes an autonomous system. (The index runs among the K′ modes that participate in a sequence J out of a repertoire of K modes, and also indicates the specific order of the sequence, i.e., the sequence starts with mode with j = 1 and terminates with j = K′. The index i runs among the dimensions of the state variables).
Figure 4.
Simulation of the functional architecture generating the word ‘flow’.
Panel A shows the generation of the word ‘flow’ and the operational signals involved. The word is repetitively generated after a short transient (black solid line). Four principle functional modes are used, one for each character (associated with solid blue, green, magenta and cyan lines, respectively), plus two auxiliary ones at the sequence's beginning and end (dotted dark and light brown lines, respectively). From top to bottom: three repetitions of the word in the handwriting workspace (the plane x-y), the output trajectory in the 3-dimensional functional phase space spanned by state variables x, y and z, followed by their time series, and the time series of the slow (WTA competition coefficients {|ξj|}) and the instantaneous (δy,z ‘kicks’, light and dark red, respectively) operational signals. The {ξj} of the modes that do not participate in the word always have a value close to zero (red line). Panel B shows the feedback loop from the output dynamics to the slow sequential one. From top to bottom: time series of the inhibitory feedback functions Fjinh, the slow feedback integrating variables vj, the absolute values of the (fast) ‘switching’ variables λj, and the WTA competition parameters Cj and Lj. These quantities vary on the time scale of a whole word (except for Fjinh that varies at the time scale of a movement cycle), even if they also contain fast changes during their evolution. The parameter values for this simulation were as follows: noise standard deviation was s = 0.001, while kjinh = [6,12,5,5,2.67,6] and kjexc = [12], [11], [10], [9], [8], [7] for each mode in the sequence, respectively (only these parameters that have to be manually set prior to a simulation). The initial conditions for the functional mode dynamics were x0 = 0, y0 = 0.1, and z0 = −0.1, while those of {νj}, {λj} and {ξj} where chosen randomly from a uniform distribution in the interval [0,1] for {νj} and {λj}, and [0,1/K] for {ξj}.
Figure 5.
The figure shows the architecture's output generating the character ‘w’ with a different movement amplitude at each column: from left to right, the radius of the cylindrical manifold is the default one (r = 1), two times larger (r = 2) and its half (r = 0.5). From top to bottom: the architecture's output in the handwriting workspace (the plane x-y), and the state variables' time series (x, y, and z). The duration as well as the profile of each stroke's time series is almost identical for all values of the movement amplitude (the isochrony principle).
Figure 6.
From top to bottom: means and standard deviations of y(t) (denoted as yµ(t) and ys(t)) of dy(t)/dt ((dy/dt)µ(t), and (dy/dt)s(t)), of δy(t) (δyµ(t) and δys(t)), of z(t) (z µ(t) and z s(t)) of dz(t)/dt ((dz/dt)µ(t), and (dz/dt)s(t)), of δz(t) (δzµ(t) and δzs(t)), and of {ξj(t)} (ξj µ(t) and ξj s(t)). Means are plotted in blue and standard deviations in green except for the graph of {ξj(t)} where colors correspond to different modes and where means and standard deviations are plotted with a continuous and dashed line, respectively. Grey and pink shadings focus on the segments of increased {ξj(t)} and δy,z(t) variability, respectively. Notice the strong effect of δ-‘kicks’ on the means and standard deviations of the state variables' rates of change ((dy/dt)µ(t), dz/dt)µ(t) and (dy/dt)s(t), (dz/dt)s(t)). Instead, the variability of {ξj(t)} (ξjs(t)) has a much weaker effect on (dy/dt)s(t) and (dz/dt)s(t), and cannot be unambiguously distinguished from the rest of the (dy/dt)s(t) and (dz/dt)s(t) variability.
Figure 7.
From top to bottom: means and standard deviations of y (denoted as yµ and ys), of dy/dt ((dy/dt)µ and (dy/dt)s), of δy (δyµ and δys), of z (z µ and z s), of dz/dt ((dz/dt)µ and (dz/dt)s), of δz (δzµ and δzs), and of {ξj} (ξj µ and ξj s). Colors, shadings and line styles are similar as in Figure 6. The effect of δ-‘kicks’ on the architecture's output is as evident as in Figure 6 (notice also that (dy/dt)s and (dz/dt)s are almost identical to δys and δzs, respectively, in the segments with a δ-‘kick’). The variability of {ξj} (ξjs) that signals mode transitions, now has a significant effect on standard deviations of dy/dt and dz/dt that approximate the phase flow. This effect cannot be identified unambiguously in the variability of the trajectory in the phase space (ys and zs). At the first transition, the δ-‘kick’ variability follows that of the {ξj}, and their effects are easily separable. Instead, at the second transition, the mean of the {ξj} modulates the standard deviation of δz and thereby the one of dz/dt as well (because of their overlapping in the data set). At the third transition no δ-‘kick’ is involved; however, there is still a significant increase in (dy/dt)s and (dz/dt)s due to the increase in ξj s.