Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Phase diagram

We study the stable phases of an attractor neural network model, with binary units, for hippocampal place cells encoding one-dimensional (1D) or 2D spatial maps or environments. Different maps correspond to random allocations (permutations) of the place fields. Based on replica calculations we show that, below critical levels for the noise in the neural response and for the number of environments, the network activity is spatially localized in one environment. For high noise and loads the network activity extends over space, either uniformly or with spatial heterogeneities due to the crosstalk between the maps, and memory of environments is lost. Remarkably the spatially localized regime is very robust against the neural noise until it reaches its critical level. Numerical simulations are in excellent quantitative agreement with our theoretical predictions.

Figure 1

Example of remapping of the place field centers of $N=6$ neurons (denoted by indices 1,...,6) in two different one-dimensional (1D) environments with periodic boundary conditions and $w=\frac{2}{6}$. Place fields in each environment are represented by colored dashed lines, place field centers are denoted by letters a,...,f.

Figure 2

Average activity $\rho \left(x\right)$ in dimension $D=1$ in the paramagnetic phase (a) and in the clump phase ((b): temperature $T=0$; (c): temperature $T=0.0073$) for $\alpha =0$, computed with $M=2000$ bins of discretization.

Figure 3

The paramagnetic (PM) phase is stable in the upper part of the $(\alpha ,T)$ plane, defined by $T>T_{\mathrm{PM}}\left(\alpha \right)$. The spin glass (SG) phase exists below this line; replica symmetry is broken everywhere in the $T<T_{\mathrm{PM}}\left(\alpha \right)$ region.

Figure 4

Highest temperature at which the clump exists, $T_{\mathrm{CL}}$, as a function of the number $M$ of discretization bins for three values of $w$. The average activity is $f=0.1$ and the load vanishes, $\alpha =0$.

Figure 5

Two-dimensional clump of activity $\rho (x,y)$ for a single environment ($\alpha =0$) at temperature $T=0.0055$ computed with $M=400$.

Figure 6

Effect of the load $\alpha$ on the clump: average activity $\rho \left(x\right)$ in dimension $D=1$ in the clump phase at temperature $T=0.004$ for $\alpha =0$ (left) and $\alpha =0.02$ (right).

Figure 7

Domain of stability the clump phase, computed with $M=200$ bins. Longitudinal and replicon instability lines correspond to, respectively, the full and dashed lines. Due to the computational effort required for the calculation of the replicon eigenvalues, only a few points (black dots) were computed.

Figure 8

Phase diagram in the $(\alpha ,T)$ plane in $D=1$. Thick lines: transitions between phases. Thin dashed lines reproduce stability regions described above. Critical lines were computed with $M=200$.

Figure 9

$q$ as a function of $T$ for fixed $\alpha$: $\alpha =0$ (solid line), $\alpha =0.01$ (dashed line), and $\alpha =0.015$ (dots), computed with $M=1000$. A discontinuity is observed at the clump-paramagnetic transition.

Figure 10

$q$ as a function of $\alpha$ for fixed temperature: $T=0.002$ (solid line) and $T=0.004$ (dashed line), computed with $M=1000$. A discontinuity is observed at the clump-glass transition.

Figure 11

Average energy for the unidimensional model with a single environment and for increasing sizes $N$. For each size, we plot the average energy obtained after thermalization for $10N$ Monte Carlo steps starting from the uniform and from the clump configurations. Each point is averaged over 1000 simulations.

Figure 12

Correlation $C\left(d\right)$ between spins at distance $d$ (35) at low (left) and high (right) temperatures, and for various sizes $N$. (a) $T=0.004$, (b) $T=0.01$. Note the difference of logarithmic scale on the $y$ axis between the two panels.

Figure 13

Chemical potential $\frac{1}{N}{\sum }_j{J}_{ij}^0{\sigma }_j$ as a function of $x$ for $\alpha =0.01$ and $T=0.004$: theoretical prediction $\mu \left(x\right)$ (solid line) and result of simulation for one set of environments (dashed line) with $N=10000$, averaged over 100 rounds of $10N$ steps each.

Figure 14

Contribution $h_i$ of environments $\ell \ge 1$ to the local fields as a function of $x=i/N-0.5$ for the same model as in Fig. 13. Inset: histogram of $h_i$ (rectangles) compared to the Gaussian distribution of mean $fLw$ and standard deviation $\sqrt{\alpha r}$ (solid line). The value $\sqrt{\alpha r}\simeq 6.98\times {10}^{-3}$ was obtained from the resolution of (28).

Figure 15

Density of energy in the environment coherent with the clump for constant $\alpha =0.01$ (same realization of the disorder): results of Monte Carlo simulations for $N=2000$ (circles) and $N=5000$ (triangles) with error bars, compared to theoretical result computed with $M=1000$ (line).

Figure 16

Density of energy in the environment coherent with the clump for constant $T=0.004$: results of Monte Carlo simulations for $N=2000$ (circles) and $N=5000$ (triangles) with error bars, compared to theoretical result computed with $M=1000$ (line).

Figure 17

Monte Carlo simulations around the clump-glass transition for $T=0.004$: fraction of simulations found in the glassy phase after 100 rounds of $10N$ steps, as a function of $\alpha$ and for different $N$, with error bars. For each point the fraction was calculated from 50 simulations, half of which were started in a clump configuration and the other half in a uniform configuration.

Figure 18

Influence of $w$ on the clump phase: $T_{\mathrm{CL}}$ (top) and ${\alpha }_{\mathrm{CL}}$ (bottom) as a function of $w$, for different fixed values of $f$. Note the maximum around $w\sim f$ in the latter graph. Computations were done with $M=1000$. The numerical error is $\delta {\alpha }_{\mathrm{CL}}\sim 0.005$.

Figure 19

Influence of $w$ on the first-order transitions: $T_c$ (top) and ${\alpha }_g$ (bottom) as a function of $w$, for different fixed values of $f$. Computations were done with $M=1000$.

Figure 20

Influence of $f$ on the clump phase: $T_{\mathrm{CL}}$ (top) and ${\alpha }_{\mathrm{CL}}$ (bottom) as a function of $f$, for different fixed values of $w$. Note the maximum around $f\sim w$ in the latter graph. Computations were done with $M=1000$. The numerical error is $\delta {\alpha }_{\mathrm{CL}}\sim 0.005$.

Figure 21

Influence of $f$ on the first-order transitions: $T_c$ (top) and ${\alpha }_g$ (bottom) as a function of $f$, for different fixed values of $w$. Computations were done with $M=1000$.

Figure 22

Effect of partial activity: Influence of the fraction $c$ of active cells on the clump domain: $T_c$ (top) and ${\alpha }_g$ (bottom) as a function of $c$, for different fixed values of $f$ and $w$. Computations were done with $M=200$.

Figure 23

Solid line: longitudinal stability region of the clump phase for $D=2$. The $D=1$ case is shown in thin dashed line for comparison.

Figure 24

Two examples of pairings between $2k$ points: ${\mathcal{P}}_A$ (left) and ${\mathcal{P}}_B$ (right).