Bimodular continuous attractor neural networks with static and moving stimuli

We investigated the dynamical behaviors of bimodular continuous attractor neural networks, each processing a modality of sensory input and interacting with each other. We found that when bumps coexist in both modules, the position of each bump is shifted towards the other input when the intermodular couplings are excitatory and is shifted away when inhibitory. When one intermodular coupling is excitatory while another is moderately inhibitory, temporally modulated population spikes can be generated. On further increase of the inhibitory coupling, momentary spikes will emerge. In the regime of bump coexistence, bump heights are primarily strengthened by excitatory intermodular couplings, but there is a lesser weakening effect due to a bump being displaced from the direct input. When bimodular networks serve as decoders of multisensory integration, we extend the Bayesian framework to show that excitatory and inhibitory couplings encode attractive and repulsive priors, respectively. At low disparity, the bump positions decode the posterior means in the Bayesian framework, whereas at high disparity, multiple steady states exist. In the regime of multiple steady states, the less stable state can be accessed if the input causing the more stable state arrives after a sufficiently long delay. When one input is moving, the bump in the corresponding module is pinned when the moving stimulus is weak, unpinned at intermediate stimulus strength, and tracks the input at strong stimulus strength, and the stimulus strengths for these transitions increase with the velocity of the moving stimulus. These results are important to understanding multisensory integration of static and dynamic stimuli.

Figure 1

The responses of a unimodular CANN when the external input is only applied during 0 to $2{\tau }_s$. The responses gradually fade after the input is removed. Light blue lines indicate the input positions ($\pi$).

Figure 2

The bimodular CANNs architecture.

Figure 3

The firing rates (responses) of bimodular CANNs when various intermodular couplings are imposed. In each inset, the top (bottom) plot shows the population responses of first (second) network module. External inputs $I_{01}=I_{02}=0.7$. Other parameters are ${\omega }_{11}={\omega }_{22}=1,a=0.5,k=1.1$. $I_{1\text{ext}}$ is fixed at position $3\pi /2$, and $I_{2\text{ext}}$ is fixed at position $\pi$. Light blue lines indicate trajectories of inputs.

Figure 4

The displacement effect on bump height. The bumps in both modules are displaced by intermodular interaction. The intermodular couplings: ${\omega }_{12}=0.1, {\omega }_{21}=-0.1$. Other parameters: ${\omega }_{11}={\omega }_{22}=1,a=0.5,k=1.1$. $R_1$ denotes responses of module 1, and $R_2$ denotes responses of module 2. The magnitudes of $I_{01}$ and $I_{02}$ are 0.5. Inputs 1 and 2 are applied at $1.5\pi$ and $\pi$, respectively. The displacement between input position (blue lines) and peak position of response (red lines) of each module is denoted and marked by ${\mathrm{\Delta }}_1\left({\mathrm{\Delta }}_2\right)$.

Figure 5

The center-of-mass position of the responses in module 1 vs the input position of module 1 at different weak intermodular couplings. Different colors indicate different $I_{01}$ of Input 1. Input 2 is fixed at position 0 and $I_{02}=0.7$. ${\omega }_{11}={\omega }_{22}=1,a=0.5,k=1.1$. Solid lines: Steady states reached from the all-quiet initial condition. Dashed lines: Steady states reached from other initial conditions. Values of $I_{01}$ are indicated in the legend. (a) Intermodular couplings ${\omega }_{21}={\omega }_{12}=0.1$. (b) Intermodular couplings ${\omega }_{21}=0.1,{\omega }_{12}=-0.1$. (c) Intermodular couplings ${\omega }_{21}=-0.1,{\omega }_{12}=0.1$. (d) Intermodular couplings ${\omega }_{21}=-0.1,{\omega }_{12}=-0.1$.

Figure 6

Module 1 Perception Bias as a function of disparity. Different colors indicate different $I_{01}$ of Input 1. Input 2 is fixed at position 0 and $I_{02}=0.7$. ${\omega }_{11}={\omega }_{22}=1,a=0.5,k=1.1$. Solid lines: Steady states reached from the all-quiet initial condition. Dashed lines: Steady states reached from other initial conditions. Values of $I_{01}$ are indicated in the legend. (a) Intermodular couplings ${\omega }_{21}={\omega }_{12}=0.1$. (b) Intermodular couplings ${\omega }_{21}=0.1,{\omega }_{12}=-0.1$. (c) Intermodular couplings ${\omega }_{21}=-0.1,{\omega }_{12}=0.1$. (d) Intermodular couplings ${\omega }_{21}=-0.1,{\omega }_{12}=-0.1$.

Figure 7

The time evolution of the center-of-mass position of module 1 in a bimodular network with ${\omega }_{21}={\omega }_{12}=0.1, I_{01}=0.4, I_{02}=0.7, {\omega }_{11}={\omega }_{22}=1,a=0.5,k=1.1, z_1=0.9\pi$, and $z_2=0$, lying in the regime with two steady states in Fig. 5, and $x_1=0.734\pi$ and $0.893\pi$. Input 1 is switched on at $t=0$, and input 2 is switched on at $t=10$, 15, 20, and 25 ms, as indicated by different colors.

Figure 8

Behavior of bimodular CANNs under different weak intermodular couplings and stimuli, one static and one moving. $I_{1\text{ext}}$ is static and located at position $\pi$. $I_{2\text{ext}}$ is a moving stimulus, moving velocity $v_2=0.01$ rad/ms. Light blue lines indicate the inputs' trajectories. $k=1.1$ and $I_{01}=I_{02}=0.7$ except $I_{01}=1.4$ in (c). In each of (a), (b), and (c), the top panel shows the sketches of the network structure, with $M1$ and $M2$ denoting modules 1 and 2, respectively. The middle panel shows the firing rates (responses) of module 1, and the bottom panel shows the firing rates (responses) of module 2.

Figure 9

The phase diagrams of the dynamical behaviors in module $m$ with moving stimulus [(a), (d), and (h)] and network behaviors at selected locations. The static input $I_{1\text{ext}}$ is fixed at the amplitude of 0.7, applied at position $\pi$, and the amplitudes of all couplings are fixed at 0.1. The trajectories of input 2 are indicated by light blue lines. (b),(c) The firing rates (responses) of the bimodular CANN in (a) when $I_{2\text{ext}}$ moves at a speed of 0.3 rad/ms, and the magnitude of $I_{2\text{ext}}$ is 1 and 3, respectively. (e), (f), and (g) The firing rates (responses) of the bimodular CANN in (d) when $I_{2\text{ext}}$ moves at a speed of 0.3 rad/ms, and the magnitude of $I_{2\text{ext}}$ is 0.4, 2, and 6, respectively. (i) and (j) The firing rates (responses) of the bimodular CANN in (h) when $I_{2\text{ext}}$ moves at a speed of 0.3 rad/ms, and the magnitude of $I_{2\text{ext}}$ is 1 and 6, respectively.

Figure 10

The firing rates (responses) of the traditional model under different weak intermodular couplings and stimuli, one static and one moving, which are indicated by light blue lines. (a) The same parameter settings as Fig. 8. (b) The same parameter settings as Fig. 8. (c) The same parameter settings as Fig. 8.

Figure 11

The firing rates (responses) of the traditional model under different weak intermodular couplings and stimuli, one static and one moving, indicated by light blue lines. The figures with the same parameter settings are indicated in the title of each panel.