Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Collective motion of the activity

The dynamics of a neural model for hippocampal place cells storing spatial maps is studied. In the absence of external input, depending on the number of cells and on the values of control parameters (number of environments stored, level of neural noise, average level of activity, connectivity of place cells), a “clump” of spatially localized activity can diffuse or remains pinned due to crosstalk between the environments. In the single-environment case, the macroscopic coefficient of diffusion of the clump and its effective mobility are calculated analytically from first principles and corroborated by numerical simulations. In the multienvironment case the heights and the widths of the pinning barriers are analytically characterized with the replica method; diffusion within one map is then in competition with transitions between different maps. Possible mechanisms enhancing mobility are proposed and tested.

Figure 1

Sketch of the phase diagram in the plane of neural noise, $T$, and number of environments per neuron, $\alpha$. Thick solid lines: transitions between phases. Thin dashed lines: stability region of each phase against fluctuations. Insets show the corresponding activity profiles in the 2D model (averaged over one round of Monte Carlo simulations after thermalization). In the clump phase we represent the same activity profile in the retrieved environment (top) and in another stored environment (bottom). See Ref. [25] (Fig. 8) for more quantitative details.

Figure 2

Sketches of the clump of neural activity moving in space, shown at two subsequent times (only central parts are shown), in the 1D, single-environment case with $T=0.006$. The dashed lines represent the equilibrium profile ${\rho }^{*}\left(x\right)$. Full lines correspond to average densities computed at the two times under consideration, which deviate from ${\rho }^{*}$ by terms of the order of $N^{-1/2}$. The horizontal dotted lines locate $\rho =f$ and $\rho =1$. Simulations parameters: $N=2000$, activity averaged over short distance (10 spins) and time ($5N$ Monte Carlo steps).

Figure 3

Sketch of the free-energy landscape probed by the clump of neural activity (dashed curve) moving through space. Fluctuations of the free energy are of the order of $\Delta F$ and are correlated over a space scale equal to ${\ell }_b$.

Figure 4

Position $x_c$ vs time $t$ of a freely diffusing clump in dimension 1, for $\alpha =0$ and $50000$ rounds of Monte Carlo simulation with $N=333$ neurons, and noise $T=0.006$. Time unit = one round of $20N$ steps.

Figure 5

Trajectory of a freely diffusing clump in dimension 2, for $\alpha =0$ and $50000$ rounds of Monte Carlo simulation with $N=32\times 32$ neurons, and noise $T=0.005$. Time unit = one round of $20N$ steps.

Figure 6

Diffusion of the clump in the single-environment case ($\alpha =0$) and 1D space. The theoretical prediction for the diffusion constant, $D_0$, given by Eq. (31), is plotted as a function of $\frac{1}{N}$ for $T=0.005$ (full lines) and $T=0.006$ (dashed lines) and compared to the results of Monte Carlo simulations $D_{\text{sim}}$ (after correction of the binning effect). The agreement with the analytical prediction (done in the $N\to \infty$ limit) improves as $N$ increases. This also explains why the discrepancy is larger than error bars for smaller $N$. Therefore, simulations corroborate well the theoretical analysis, and the diffusion properties of the clump can be understood analytically in the single-environment case. Simulation time: 1000 rounds of $100N$ steps. Depending on the computational cost, each point is averaged over a number of simulations ranging from 5 (for large $N$) to 100.

Figure 7

Standard deviation $\beta \sqrt{V}$ of the free energy (in units of the temperature and divided by $\sqrt{N}$) as a function of the load $\alpha$ for fixed temperature $T$. Lines end at the clump instability limit.

Figure 8

Contour lines of constant $N_c$ in the phase diagrams of the 1D (top) and 2D (bottom) models. In one dimension, for a given $N_c$, the area of the diagram above the contour line corresponds to free diffusion, while in the area below the diffusion is activated. In two dimensions, this distinction is less clear due to the possible by-passing of free-energy barriers (see text).

Figure 9

Covariance $W(x-y)$ of the free energies of the clump centered on positions $x$ and $y$, normalized by $V$. Results are shown for dimension 1, with $T=0.006$, $\alpha =0.01$ (full line), and in dimension 2 with $T=0.004$, $\alpha =0.002$ (dashed line).

Figure 10

Logarithm of the diffusion constant $D$ as a function of $\sqrt{N}$ with constant $L+1=2$, measured in Monte Carlo simulations in both dimensions 1 and 2. For sufficiently large $N$, ${log}_{10}\left(D\right)$ seems to decrease linearly with $\sqrt{N}$. The simulations length depends on the frequency of transitions: typically, of the order of $10–{10}^2$ rounds for $\sqrt{N}=26$ and 1000 rounds for $\sqrt{N}>35$. Depending on the computational cost, each point is averaged over a number of simulations ranging from 5 (for large $N$) to 100.

Figure 11

Contour lines of constant $N_c$ in the 1D phase diagram for different values of $w$, $f$. Note the quantitative change in the $T$ axis. The qualitative aspect is remarkably preserved.

Figure 12

Example of a transition observed in a Monte Carlo simulation with $N=1000$ neurons, $L+1=2$ environments, and $T=0.006$. Neural configurations $\sigma$ are shown at different times (black dots correspond to active neurons). Both panels show the same data, with the difference that neurons are ordered according to their place field centers ${\pi }^1\left(i\right)$ in environment 1 (top) and ${\pi }^2\left(i\right)$ in environment 2 (bottom). The transition takes place around time $t\simeq 15$ (time unit: one round of $N$ steps).

Figure 13

Evolution of $E_{\ell }\equiv {\sum }_{i<j}{J}_{ij}^{\ell }{\sigma }_i{\sigma }_j$, for the same transition event as in Fig. 12. $E_{\ell }$ is the contribution of environment $\ell$ to the logarithm of the probability of the neural configuration $\sigma$; see (2). The crossing of $E_1$ and $E_2$ defines the transition between the two maps, as well as the intermediary state, where the activity is weakly localized in both maps.

Figure 14

Rate of transitions to other environments as a function of $N$ for one realization of $L+1=2$ 1D environments and $T=0.006$. Each point is averaged over 10 simulations of 1000 MC rounds. Time unit: 1 round of $N$ steps. The decay of the rate is consistent with an exponentially decreasing function of $N$, hence with Arrhenius's law and the existence of free-energy barriers proportional to $N$.

Figure 15

Contour lines in the $(\alpha ,T)$ plane corresponding to a fixed crossover size, $N_c$, for the 1D case with $c=0.5$.

Figure 16

Effect of partial activity on the theoretical free-energy barriers $\beta \sqrt{V}$ (top), on the diffusion constant $D$ (bottom, left) and on the rate of transitions per round (bottom, right). Results correspond to the 1D case, $T=0.003$, $\alpha =0.003$, $N=333$. The dashed line indicates the limit of stability of the clump. The simulations length depends on the frequency of transitions: typically 1000 rounds for $1-c=0$ and of the order of ${10}^2$ rounds for $1-c=0.6$. One round = $100N$ steps. Each point is averaged over 100 simulations.

Figure 17

Effect of partial activity on the diffusion constant $D$ in the 2D case, with $T=0.002$, $\alpha =0.001$, $N=45\times 45$ units. The clump phase is not stable anymore when $1-c$ exceeds $\simeq 0.6$. The simulation length depends on the frequency of transitions: typically, of the order of ${10}^3$ rounds for $1-c=0$ and of the order of ${10}^2$ rounds for $1-c=0.6$. 1 round = $100N$ steps. Each point is averaged over 30 simulations.

Figure 18

Mobility of the clump in response to an external force, in the single-environment ($\alpha =0$), 1D case. The theoretical prediction for the effective mobility, ${\mu }_{0,\text{th}}$, computed from Eq. (67), is plotted as a function of $\frac{1}{N}$ for $T=0.005$ (dashed lines) and $T=0.006$ (full lines) and compared to the results of Monte Carlo simulations ${\mu }_{0\text{sim}}$. The agreement with the analytical prediction [which neglects terms smaller than $O\left(\frac{1}{N}\right)$] improves as $N$ increases. Each point is averaged over 10 simulations, in which the clump, initially at location $x=0$, had moved over four space bins (the environment is covered by 11 bins).

Figure 19

Velocity of the clump under a force $A$. Top: dimension one, $T=0.006$, $N=1000$ (the clump is not stable for larger $A$ as indicated by the dashed line). Bottom: dimension two, $T=0.005$, $N=32\times 32$. Simulation time: 1000 rounds for $\alpha =0$; around ${10}^3–{10}^4$ rounds for $\alpha >0$. Each point is averaged over 10 simulations; one round = $20N$ steps.

Figure 20

Bypassing of barriers in two dimensions. Top: Trajectory of the clump in the $(x,y)$ plane under the effect of a force oriented rightward along the $x$ axis, indicated by an arrow. Parameters are $T=0.004$, $\alpha =0.001$, $N=32\times 32$, $A=0.02$, simulation time = $3.5\times {10}^5$ rounds of $20N$ steps. Bottom: Resulting contour plot of $\widehat{F}\equiv -log{\tau }_{\mathrm{tot}}(x,y)$, where ${\tau }_{\mathrm{tot}}(x,y)$ is the total time spent in position $(x,y)$ after incorporation of periodic boundary conditions. $\widehat{F}$ is an estimate of the free-energy landscape: deep local minima, hills and valleys appear; see gray-level scale on the right side.

Figure 21

Evolution of the overlap with the retrieved pattern as a function of time, during two Monte Carlo simulations initialized in the clump phase in the same environment, at a position different from $x_0$. $N=1000$, $T=0.006$, $\alpha =0.01$, $d_0=0.05$, time unit = 1 round of $20N$ steps.

Figure 22

Average retrieval time in Monte Carlo simulations as a function of $h$ (top) and $d_0$ (bottom). Each point is averaged over 10 simulations. $N=1000$, $T=0.006$, $\alpha =0.01$, time unit = 1 round of $20N$ steps. In the top panel, $d_0=0.05$.

Figure 23

Average retrieval time in Monte Carlo simulations as a function of $h_J$, with two different initial conditions: a system in the paramagnetic phase (triangles) or in the clump phase in another environment (circles). Each point is averaged over 100 simulations. $N=1000$, $T=0.006$, $\alpha =0.003$, time unit = one round of $20N$ steps.

Figure 24

Adaptation: $D$ (top) and frequency of transitions (bottom) as a function of the adaptation's intensity $h_{\text{adapt}}$ measured in a Monte Carlo in dimension 1 with $N=333$ spins, $\alpha =0.003$, $T=0.004$, and various values of ${\tau }_{\mathrm{adapt}}$ (time unit: 1 round of $20N$ steps). The clump is not stable for stronger $h_{\text{adapt}}$. Depending on the frequency of transitions, simulation durations range from $\sim 10$ to 1000 rounds; each point is averaged over 100 simulations. The estimated error on $D$ varies between $5\times {10}^{-5}$ to a few ${10}^{-3}$ when $h_{\text{adapt}}$ increases from 0 to 0.001.

Figure 25

Simulations of the case $f\left(t\right)=f+\delta fsin(t/\tau )$: $D$ as a function of $\delta f$ measured in a Monte Carlo in dimension 1 with $N=1000$ spins, $\alpha =0.003$, $T=0.005$, $w=0.05$, and various values of $\tau$ (time unit: $100N$ steps). The clump is not stable for stronger $\delta f$. For $\tau =10$ and $\tau =100$, each point is averaged over 100 simulations of length varying from a few tens of rounds to 1000 rounds depending on the frequency of transitions. For $\tau =1000$, longer simulations were necessary in order to cover several periods of $f\left(t\right)$; each point is thus averaged over 10 simulations of duration up to 25 000 rounds.

Figure 26

Effect of asymmetric random dilution of synapses on $D$ (top) and on the frequency of transitions to other environments (bottom): Monte Carlo simulations in dimension 1 with $N=333$ spins, $\alpha =0.003$, $T=0.005$. The clump is not stable for stronger dilution. Depending on the frequency of transitions, the simulations length varies between 1000 rounds and a few rounds. Each point is averaged over 1000 simulations. The estimated error on $D$ varies between $5\times {10}^{-6}$ to ${10}^{-4}$ when ${\delta }_{\mathrm{dil}}$ increases from 0 to 0.6.

Figure 27

Overlap $q_{12}$ between two groups of replicas centered respectively on positions $x$ and $y$, in dimension 1 with $T=0.006$, $\alpha =0.01$ (full line) and in dimension 2 with $T=0.004$, $\alpha =0.002$ (dashed line). Dotted lines indicate $q_{12}=f$ and $q_{12}=f^2$.