Dynamic Organization of Hierarchical Memories

<div><p>In the brain, external objects are categorized in a hierarchical way. Although it is widely accepted that objects are represented as static attractors in neural state space, this view does not take account interaction between intrinsic neural dynamics and external input, which is essential to understand how neural system responds to inputs. Indeed, structured spontaneous neural activity without external inputs is known to exist, and its relationship with evoked activities is discussed. Then, how categorical representation is embedded into the spontaneous and evoked activities has to be uncovered. To address this question, we studied bifurcation process with increasing input after hierarchically clustered associative memories are learned. We found a “dynamic categorization”; neural activity without input wanders globally over the state space including all memories. Then with the increase of input strength, diffuse representation of higher category exhibits transitions to focused ones specific to each object. The hierarchy of memories is embedded in the transition probability from one memory to another during the spontaneous dynamics. With increased input strength, neural activity wanders over a narrower state space including a smaller set of memories, showing more specific category or memory corresponding to the applied input. Moreover, such coarse-to-fine transitions are also observed temporally during transient process under constant input, which agrees with experimental findings in the temporal cortex. These results suggest the hierarchy emerging through interaction with an external input underlies hierarchy during transient process, as well as in the spontaneous activity.</p></div

The neural dynamics after 40 learning steps in the response (R) regime.

<p><b>A.</b> The time series of the neural activities shown by the overlap with the 1st, 5th, and 30th targets in the absence and presence of the 1st (red), 5th (green), and 30th (blue) inputs (shown by the colored bars above the plot) for . <b>B.</b> The time-averaged overlaps with the learned targets as a function of (squares). The overlaps with the targets and inputs averaged over the 100 networks are shown as the solid and dashed lines, respectively. <b>C.</b> The distributions of the overlaps of the spontaneous activity with the targets. The black line represents the distribution averaged over 10 overlaps with 10 random patterns as a control, and the others are distributions of the overlaps , , and using the same colors as in A. <b>D.</b> The SD of the overlap with the target for the temporal evolution (squares), and the SD of the target and random pattern averaged over the 100 networks shown as the right blue and black lines, respectively.</p

The transient neural activity before reaching the attractor.

<p>A-C) Transient dynamics of the neural activity under input 11 with <i>γ</i> = 5. The dynamics is colored before reaching the attractor in magenta and after convergence to the attractor in red. Neural activity after convergence to an attractor with a less intense strength (<i>γ</i> = 4) is plotted in gray, for reference. A) Time series of the overlap with the target pattern 11. B) Overlap profiles before and after convergence in magenta and red are plotted. Here, the former overlap profile is measured by averaging overlap after 50 unit-time transient up to the convergence point. The latter profile is measured after convergence. We also plot the profile for the neural activity with a less intense strength in gray as reference. C) The orbit of the neural activity dynamics in A is plotted, by projecting it into the two-dimensional space. The horizontal and vertical axes represent the overlaps with targets 8 and 11, respectively.</p

Phase diagram of the evoked and spontaneous dynamics and bifurcation diagram.

<p><b>A.</b> The quenched average of the overlap with the target in the evoked dynamics. <b>B.</b> The standard deviation (SD) of the overlap averaged over time and over the networks . Average values in A and B are computed over 100 networks and over . The dotted curves in A and B, plotted for reference, show the boundary between the R and NR regimes and, which are computed by the ridge of SD in B with smoothing the line. <b>C.</b> The local maxima in the time series of the overlap with the target as a function of the input strength in (i) the NR regime for and (ii) the R regime showing the bifurcations.</p

Learning process for one mapping.

<p><b>A.</b> A raster plot of the activity for and for 25 of neurons. <b>B.</b> The temporal evolution of the overlap with the target for the learning process in A.</p

Characteristics of each regime.

<p>Characteristics of each regime.</p

Bifurcation diagram for (

<p><b>) = (16,0.01) in the R regime.</b> We use the network shaped after 40 learning steps. <b>A.</b> The local maxima in the time series of the overlap with the target in the presence of the corresponding input as a function of . The overlaps with (i) the 1st (), (ii) 5th (), and (iii) 30th () targets are plotted in red, green, and blue, respectively, while the data in black represent the overlap with each input (). <b>B.</b> The number of positive Lyapunov exponents of these evoked dynamics as a function of . Lyapunov exponents are calculated from the time series according to the algorithm in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002943#pcbi.1002943-vonBremen1" target="_blank">[56]</a>.</p

The time evolution of the overlap and the matrix elements.

<p><b>A.</b> The overlaps with the target and input during the learning process (i) in the NR regime for and (ii) in the R regime for . <b>B.</b> The matrix elements , and in (i) the NR regime and (ii) the R regime with the same parameters as in A.</p

Change in the neural activity pattern against the increase of the input strength.

<p>A and B) The neural dynamics (i) without input, (ii) under the input 6 with strength <i>γ</i> = 6, (iii) with <i>γ</i> = 16, and (iv) under the input 3 with <i>γ</i> = 16, indicated by colored bars above figures in white, right blue, blue and green, respectively. A) The overlaps with the targets 6, 7, and 3 are plotted as blue, red, and green lines, respectively. Targets 6 and 7 are in same category B, while target 3 is in category A. B) The probability density distribution of the neural activity and a sample trajectory in principal component (PC) space. Dotted circles represent position of the targets. C) The distribution of the overlaps with the targets. We calculated the overlap with a target under the associated input with <i>γ</i> = 16 for 360 associations (36 associations in 10 networks). D) The temporal average overlap profiles for the spontaneous activity (gray) and for evoked activities with input strength <i>γ</i> = 6 (light blue) and 16 (dark blue). For calculating temporal average, we used overlaps over 400 unit time after 100 unit time transient. Error bars are standard deviation of neural activity over 400 unit time. In the following analysis, temporal average is calculated in this way unless otherwise mentioned. E,F) Change in overlap with increasing input is shown. The overlap with all of targets under input 6 against different strength is computed in E. As samples, the overlaps with targets 3,6, and 7 are plotted in F. G) Bifurcation diagram of the overlap with target 6 through increasing the input strength of input 6 (bottom) and the largest Lyapunov exponent (Top). We plot the overlaps at every 5 unit times over 250 unit times after transient time for each input strength.</p

Temporal structure of spontaneous activity.

<p>A) Transition probability <i>P</i>_<i>μν</i> from the <i>ν</i>-th target to the <i>μ</i>-th target. We cannot compute the probability of self-visiting <i>P</i>_<i>μμ</i>, and set at 0, because we did not distinguish continuous stay of the neural state around a target from coming in-out-in the identical target. B) Transition probability <i>P</i>_<i>ab</i> from the category b to the category a. C) Transition time <i>T</i>_<i>μν</i> from the <i>ν</i>-th to the <i>μ</i>-th target which is averaged with <i>P</i>_<i>μν</i>. White tiles indicate that there is no transition and we cannot calculate the transition time. D) Transition time Tab from the target b to a. all of values are calculated from the spontaneous activity (0Materials and Methods” for the detailed.</p