Discrete-Attractor-like Tracking in Continuous Attractor Neural Networks

Chi Chung Alan Fung (馮志聰) and Tomoki Fukai (深井朋樹)

Phys. Rev. Lett. 122, 018102 – Published 8 January 2019

Abstract

Continuous attractor neural networks generate a set of smoothly connected attractor states. In memory systems of the brain, these attractor states may represent continuous pieces of information such as spatial locations and head directions of animals. However, during the replay of previous experiences, hippocampal neurons show a discontinuous sequence in which discrete transitions of the neural state are phase locked with the slow-gamma ( $\sim 30 - 50 Hz$ ) oscillation. Here, we explore the underlying mechanisms of the discontinuous sequence generation. We find that a continuous attractor neural network has several phases depending on the interactions between external input and local inhibitory feedback. The discrete-attractor-like behavior naturally emerges in one of these phases without any discreteness assumption. We propose that the dynamics of continuous attractor neural networks is the key to generate discontinuous state changes phase locked to the brain rhythm.

Received 21 August 2018
Revised 31 October 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.018102

Physics Subject Headings (PhySH)

Neural information Neuronal network activity Neuroscience, neural computation & artificial intelligence

Physics of Living Systems

Authors & Affiliations

Chi Chung Alan Fung (馮志聰)^* and Tomoki Fukai (深井朋樹)^†

RIKEN Center for Brain Science, Hirosawa 2-1, Wako City, Saitama 351-0198, Japan

^*chichung.fung@riken.jp
^†tfukai@riken.jp

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Vol. 122, Iss. 1 — 11 January 2019

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Images

Figure 1
Basic response properties of CANNs. (a) In CANNs, neurons with similar preferred locations are mutually connected. Excitatory (Ext.) connections have a narrower range than inhibitory (Inh.) connections. This architecture allows the network to support a continuous family of stationary states. (b) Neural responses to a moving input are shown. Parameters are $\tilde{k} = 1.0$ , $a = 0.02 m$ , $τ = 2 ms$ , ${\tilde{v}}_{I} = v_{I} / a = 0.15 (ms)^{- 1}$ , and ${\tilde{A}}_{0} = 0.5$ . (c) The instantaneous speed of the center of the localized activity profile (upper), magnitude of the external input function (middle), and the gamma-band oscillation of the neural field (lower) obtained by a bandpass filter (40–60 Hz) are shown (red curves) for the entire duration of moving input. Dotted lines indicate the troughs of gamma oscillations. Parameters are the same as those in (b). (d) The location of a rat, and (e) movements across 20-ms time frames with 5-ms increments and the slow-gamma power were decoded from hippocampal activity during replay. These panels were modified from Ref. [8] with reprint permission.
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Figure 2
Phase-plane diagrams for different parameter settings. Here, $\tilde{k} = 1.0$ and $\tilde{A} = 0.5$ . (a) Phase-plane diagram shows three fixed-point solutions for ${\tilde{v}}_{I} = 0.07$ . Dashed and dot-dashed curves indicate $\tilde{s}$ nullcline and $\tilde{u}$ nullcline, respectively. (b) Phase-plane diagram gives an unstable fixed point for ${\tilde{v}}_{I} = 0.1$ . (c) Phase-plane diagram gives a stable fixed point for ${\tilde{v}}_{I} = 0.4$ . (d) Phase-plane diagram yields an unstable fixed point for ${\tilde{v}}_{I} = 1.2$ . Arrows are visualized tendencies of nearby regions. Stable (circles) and unstable (squares) fixed points are indicated.
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Figure 3
Parameter dependence of localized activity profiles. (a) ${\tilde{u}}_{0}^{*}$ of the fixed-point solutions to Eqs. (3) and (4). (b) The value of ${\tilde{s}}^{*}$ in fixed-point solutions. In (a) and (b), only the fixed-point solutions with the smallest ${\tilde{s}}^{*}$ are presented if multiple fixed-point solutions exist. (c) Real part of the eigenvalues of the linearized system around the fixed-point solutions shown in (a) and (b). Dashed curves in (c) and (f) show contours on which the real part vanishes. (d) ${⟨ \max_{x} \tilde{u} (x, t) ⟩}_{t}$ measured from simulations of Eq. (1). (e) ${⟨ \tilde{s} (t) ⟩}_{t}$ measured from the simulations. (f) Standard deviation of the measured $\tilde{s} (t)$ . The circles labeled by $P$ show the parameter values used in Fig. 4.
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Figure 4
Simulations with a nonoscillatory input. (a) Activity state $\tilde{u} (x, t)$ evolves with a nonoscillatory input, i.e., $f = 0$ . Values of other parameters are the same as those in Fig. 1 and labeled by $P$ in Fig. 3. (b) The instantaneous speed of the activity state profile presented in (a). (c) The gamma-band component of $\max_{x} \tilde{u} (x, t)$ shown in (a). Dotted lines show the aligned troughs of the gamma-band trace in (c).
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Figure 5
Robustness of oscillatory responses against changes in ${\tilde{v}}_{I}$ , ${\tilde{A}}_{0}$ , and $f$ . The parameter values used in Fig. 4 (circle) and Figs. 1 and 1 (triangles) are indicated. (a) $[\max_{t, x} \tilde{u} (x, t) - \min_{t} \max_{x} \tilde{u} (x, t)]$ versus ${\tilde{A}}_{0}$ is shown during the tracking of a constant input moving at speed ${\tilde{v}}_{I}$ . (b) $[\max_{t, x} \tilde{u} (x, t) - \min_{t} \max_{x} \tilde{u} (x, t)]$ versus $f$ is shown during the tracking of an oscillatory input moving at speed ${\tilde{v}}_{I}$ and oscillating at frequency $f$ with magnitude ${\tilde{A}}_{0} = 0.5$ . (c) Correlations between $\tilde{v} (t)$ and the gamma-band component of $\max_{x} \tilde{u} (x, t)$ are presented during tracking of an oscillatory input moving at speed ${\tilde{v}}_{I}$ and oscillating at frequency $f$ . Parameter values are $\tilde{k} = 1.0$ and $a = 0.02 [m]$ .
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