Distributed dynamical computation in neural circuits with propagating coherent activity patterns

Abstract

Activity in neural circuits is spatiotemporally organized. Its spatial organization consists of multiple, localized coherent patterns, or patchy clusters. These patterns propagate across the circuits over time. This type of collective behavior has ubiquitously been observed, both in spontaneous activity and evoked responses; its function, however, has remained unclear. We construct a spatially extended, spiking neural circuit that generates emergent spatiotemporal activity patterns, thereby capturing some of the complexities of the patterns observed empirically. We elucidate what kind of fundamental function these patterns can serve by showing how they process information. As self-sustained objects, localized coherent patterns can signal information by propagating across the neural circuit. Computational operations occur when these emergent patterns interact, or collide with each other. The ongoing behaviors of these patterns naturally embody both distributed, parallel computation and cascaded logical operations. Such distributed computations enable the system to work in an inherently flexible and efficient way. Our work leads us to propose that propagating coherent activity patterns are the underlying primitives with which neural circuits carry out distributed dynamical computation.

Figure 1. Snapshots of three different types of spatiotemporal patterns distinguished in the circuit model.

Each neuron is oscillatory with a frequency of 11Hz. Black dots indicate the coordinates where neurons are firing. (A) Type 1 (with parameters , and ): spatially localized patterns that lightly jitter around; (B) Type 2 (with parameters and ): localized coherent patterns with long-range movements and complicated interactions. Each pattern is labeled by a distinct letter. At this time moment, the system has five spatially localized structures labeled from a to f, of which the center-of-mass positions are respectively: (5.1, 74.2), (39.7, 64.2), (9.3, 50.5), (27.3, 19.7), (15.1, 9.9), (67.2, 20.8). (C) Type 3 (with parameters , and ): localized coherent structures with regular motion and regular overall features.

Figure 2. Phase diagram of spatiotemporal activity patterns in the parameter space.

The region B circled by the dark line is the region in which the system generates the Type 2 patterns, the region A for Type 1 patterns, and the region C for Type 3 patterns.

Figure 3. Statistical properties of the propagating coherent spiking patterns.

(A) The distribution of the propagating speeds of the dynamical patterns; (B) the distribution of the patterns' sizes.

Figure 4. Log-log plot of mean-squared-displacement (MSD) as a function of time increment for different collective motions.

The red dots, representing the coherent patterns of Type 2, show a clear straight-line part in the log-log plot, fitting to an exponent of 1.73 (dashed line). The cut-off is due to the finite-size of the circuit. For the black dots, representing Type 3 patterns, the exponent of the fitted line is 2.0. The blue dots represent random Brownian motion, corresponding to an exponent of 1.0.

Figure 5. The space-time behavior of two, moving localized coherent patterns and the dynamical logical operations based on the interaction between them.

(A) Among several other activity patterns, two of them are shown here. Different colors indicate the patterns at different time moments: blue at t1 = 620503 ms, red at t₂ = 620507 ms, and green at t₂ = 620510 ms. At t₁, Pattern a is centered at position P1 (57.8, 50.4) and Pattern b at Q1 (75.1, 32.6). Time t₂, is the moment when the collision between the two moving patterns happens. At that time, Pattern a is at P2 (54.0, 41.1) and Pattern b at Q2 (64.4, 31.6). Afterwards, at t₃ Pattern a is at P3 (47.3, 39.5) and Pattern b at Q3 (59.0, 25.9). Without a mutual collision, Pattern a would have traveled along the dashed black line and passed through P4 at time moment t₃, and Pattern b along the dashed red line and through Q4 at t₃. To see the positions P1, Q1, P3, Q3, P4, and Q4 clearly, they are also noted by red circles. (B) An illustration of interaction-based logical operations. The two filled black circle are used to represent patterns at the time moment when they collide with each other. Arrows correspond to their trajectories prior to and after the collision; dotted arrows correspond to trajectories that occur if the other signal is absent. The signal AB represents ‘A AND B’, and represents ‘A AND NOT B’. A, B, AB, BA, and are corresponding signals at the positions P1, Q1, P3, Q3, P4, and Q4.

Figure 6. The space-time behavior of several localized coherent patterns and cascaded computational operations.

(A) At time , the two coherent Patterns a and b depicted in blue collide with each other. The output is a coherent Pattern c (red) which, at time is positioned at P2, where it collides with Pattern d coming from a different direction. The dashed black lines show the trajectories of the propagating patterns. (B) Illustration of the cascaded logical operations. A, B, C, D are signals signifying the presence or absence of coherent activity patterns at time moments t₁ and t₂. Based on these signals, the logical operation occurring at t₁ is located on the green dotted line, and the one at t₂ is located on the black dotted line. The output signal from the operation at t₁ that involves A and B is a signal C, C = A AND B, which acts as the input signal for the operation happening at t₂, which also involves a signal D. The dashed arrows correspond to the situation that one of these signals is absent.

Figure 7. The space-time behavior of propagating, coherent activity patterns and corresponding parallel computations.

(A) At time moment t = 862039 ms, two collisions happen simultaneously. Pattern g collides with Pattern h, and Pattern k collides with Pattern l. The dashed lines are the trajectories of the five moving coherent patterns. (B) Illustration of interaction-based parallel logical operations. A, B, C and D are input signals. The vertical green dashed line indicates where two computations happen at the same time moment , ms. The two black filled circles represent a pair of signals involved in one logical operation, and the two red filled circles represent the signals in another one. Each of these operations can produce ‘AND’, ‘AND NOT’ functions. The dashed arrows correspond to the situation that one of these signals is absent.

Figure 8. The number of co-occurring localized, coherent spiking patterns as a function of time, with the parameters , .

Figure 9. The space-time behavior of propagating coherent patterns with and without external perturbations.

(A) The blue dots are original propagating trajectories of Pattern a (green color) and Pattern b (black color) without external perturbations. The two patterns collide when Pattern a is located at (34.35, 35.86) and Pattern b is at (22.4, 44.8) respectively. Before the collision, Pattern a is located at (38.9, 29.4) and Pattern b is located at (13.1, 39.2). The red dots are propagating trajectories of these two patterns after external perturbations. (B) The distribution of distances between the perturbed outgoing trajectories after collisions and the corresponding original ones.