Forgetting Leads to Chaos in Attractor Networks

Attractor networks are an influential theory for memory storage in brain systems. This theory has recently been challenged by the observation of strong temporal variability in neuronal recordings during memory tasks. In this work, we study a sparsely connected attractor network where memories are learned according to a Hebbian synaptic plasticity rule. After recapitulating known results for the continuous, sparsely connected Hopfield model, we investigate a model in which new memories are learned continuously and old memories are forgotten, using an online synaptic plasticity rule. We show that for a forgetting timescale that optimizes storage capacity, the qualitative features of the network’s memory retrieval dynamics are age dependent: most recent memories are retrieved as fixed-point attractors while older memories are retrieved as chaotic attractors characterized by strong heterogeneity and temporal fluctuations. Therefore, fixed-point and chaotic attractors coexist in the network phase space. The network presents a continuum of statistically distinguishable memory states, where chaotic fluctuations appear abruptly above a critical age and then increase gradually until the memory disappears. We develop a dynamical mean field theory to analyze the age-dependent dynamics and compare the theory with simulations of large networks. We compute the optimal forgetting timescale for which the number of stored memories is maximized. We found that the maximum age at which memories can be retrieved is given by an instability at which old memories destabilize and the network converges instead to a more recent one. Our numerical simulations show that a high degree of sparsity is necessary for the dynamical mean field theory to accurately predict the network capacity. To test the robustness and biological plausibility of our results, we study numerically the dynamics of a network with learning rules and transfer function inferred from in vivo data in the online learning scenario. We found that all aspects of the network’s dynamics characterized analytically in the simpler model also hold in this model. These results are highly robust to noise. Finally, our theory provides specific predictions for delay response tasks with aging memoranda. In particular, it predicts a higher degree of temporal fluctuations in retrieval states associated with older memories, and it also predicts fluctuations should be faster in older memories. Overall, our theory of attractor networks that continuously learn new information at the price of forgetting old memories can account for the observed diversity of retrieval states in the cortex, and in particular, the strong temporal fluctuations of cortical activity.

Popular Summary

Memories in the brain are learned and forgotten through experience. The popular theory of “attractor networks” proposes that a memory is stored as a specific pattern of brain activity (an attractor state) that persists over time. However, aspects of the theory are at odds with experiments: When memories are retrieved in the classical theory, neuronal activity is constant, whereas in some experiments the activity is highly irregular. Here, we show that there are roles for both constant and irregular activity in a network where new memories are learned continuously and old memories are forgotten.

In such a network, we find that most recent memories are retrieved as stationary attractors, whereas older memories are retrieved as chaotic attractors, characterized by strong heterogeneity and temporal fluctuations. The combination of strong neural fluctuations and stable encoding of memories during chaotic retrieval states qualitatively resembles the neural dynamics observed experimentally.

We develop a theory to analyze these age-dependent network dynamics. Our theory predicts that the network presents a continuum of statistically distinguishable memory states, where chaotic fluctuations appear abruptly above a critical age and then increase gradually until the memory disappears. This is confirmed using simulations of large networks.

Our theory qualitatively holds in more biologically realistic networks, and it provides testable predictions for memory experiments with aging memoranda. In particular, it predicts faster and higher degrees of temporal fluctuations in retrieval states associated with older memories.

Figure 1

Center: bifurcation diagram for the TT model in the parameter space spanned by memory load ($\alpha$) versus the learning gain ($A$). Full lines: boundaries of the four qualitatively different regions calculated using DMFT. The boundaries between fixed-point and chaotic retrieval states are computed using Eqs. (22)–(24). The capacity curve that separates chaotic nonretrieval states from retrieval states is computed using Eqs. (29) and (30). The vertical line that separates nonretrieval from retrieval states are computed using Eqs. (22) and (23). Red circles: location of the transition to chaos of retrieval states, using simulations of networks of $N={10}^7$ units and $K=2\log (N)$ connections per neuron. Blue circles: storage capacity, computed using simulations. Dashed red line: transition to chaos of the background state [see Eq. (26)]. Black dashed line: line on which (unstable) fixed-point retrieval states disappear [see Eqs. (22) and (23)]. Surrounding panels, labeled I–IV, show representative numerical simulations in the four qualitatively different regions. In each simulation, the network is initialized close to one of the stored memories. Each plot shows the overlaps of network state with this memory (blue) and other memories (green), and the activity of 10 randomly selected neurons as a function of time. The network parameters $A$ and $\alpha$ are indicated with a square at the bottom left-hand corner of the corresponding roman numbers. The network parameters for the surrounding panels are $N={10}^6$ and $K=2\log (N)$. The parameter values for $A$ and $\alpha$ for panels I–IV are, respectively, $A=0.5$, 2.5, 2.5, 2.5 and $\alpha =11/K,11/K,21/K,33/K$.

Figure 2

(a) Overlap versus memory load. Circles: overlaps computed from network simulations (average over 10 realizations). Dashed vertical line: transition to chaos [see Eqs. (22) and (24)]. Black lines: overlaps computed from DMFT [see Eqs. (22), (27), and (28)]. Red lines: overlaps in static solutions to DMFT (SMFT) [see Eqs. (22) and (23)] which are unstable beyond the transition to chaos. (b) Parameters characterizing the autocovariance function ${\mathrm{\Delta }}_{0s}$ and ${\mathrm{\Delta }}_{1s}$. As in (a), full lines represent solutions from DMFT [see Eqs. (22), (27), and (28)], while circles represent simulation results. (c) Autocovariance function for $\alpha =0.5429$ (i.e., $\alpha =15/K$) [solution of Eq. (20)]. Black, DMFT; yellow, 10 realizations of network simulations. Other parameters: $N={10}^6$, $K=2\log (N)$, and $A=5.5$.

Figure 3

Effect of connectivity sparseness on the dynamics of the TT model. (a) Overlap versus memory load, for different scalings of connection probability with network size $N$. Circles: overlap averaged time after transients and over 10 realizations. Horizontal dashed line: transition to chaos [see Eqs. (22)–(24)]. Black line: overlap computed from DMFT [see Eqs. (22), (27), (28)]. Network sizes corresponding to the scalings $c=1/N^{0.3},1/N^{0.4},1/N^{0.5},1/N^{0.6},1/N^{0.75},2\log (N)/N$ are $N={10}^5,2.5\times {10}^5,5\times {10}^5,{10}^6,{10}^6,{10}^6$, respectively. (b) Capacity versus $\log (c)/\log (N)$ [blue symbols for power-law scalings, olive symbols for the $\log (N)/N$ scaling] and transition to chaos versus $\log (c)/\log (N)$ (red circles). The capacity is estimated from network simulations as the minimal memory load for which the overlap is smaller than 0.1. Dashed lines: DMFT blue and red lines correspond to the capacity [see Eqs. (29) and (30)] and transition to chaos [see Eqs. (22)–(24)] predicted by the DMFT, respectively. Green square: capacity in the fully connected case [54] which is equal to the capacity of the Hopfield model (0.14). For each scaling, three symbols correspond to three different networks sizes: $N=\{0.15\times {10}^5,0.6\times {10}^5,1\times {10}^5\}$ for $c=1/N^{0.3}$; $N=\{0.5\times {10}^5,1.5\times {10}^5,2.5\times {10}^5\}$ for $1/N^{0.4}$; $N=\{1.5\times {10}^5,2\times {10}^5,5\times {10}^5\}$ for $1/N^{0.5}$; $N=\{2\times {10}^5,10\times {10}^5,20\times {10}^5\}$ for $1/N^{0.6}$; $N=\{5\times {10}^5,10\times {10}^5,15\times {10}^5\}$ for $1/N^{0.75}$; and $N=\{6\times {10}^5,20\times {10}^5,30\times {10}^5\}$ for $2\log (N)/N$. For each scaling, circle, square, and triangle correspond to smaller, intermediate, and larger network sizes, respectively. The learning gain is $A=5.5$.

Figure 4

(a) Bifurcation diagram for the network with forgetting, for $A=10$. Solid lines: transition to chaos [see Eqs. (22)–(24)] and maximal age at which retrieval states exist, calculated using the DMFT [see Eqs. (29) and (30)]. Dashed line: maximal age at which static retrieval solutions exist (SMFT) [see Eqs. (22) and (23)]. Red line: maximal age at which retrieval states are stable [see Eqs. (39)–(41)]. (b) Retrieval dynamics of memories of age 1, 5, and 10 for the same realization of the connectivity matrix [$N={10}^6$, $K=2\log (N)$, $A=10$, $\tau =0.5$]. These ages correspond to $s=1/K,5/K,10/K$, respectively, indicated with a square at the bottom left-hand corner of the corresponding roman numbers; see I–III in (a).

Figure 5

(a) Overlap versus memory age $s$ in the network with online learning. Green line (green circles): overlap with a memory of age $s$ after the network is initialized close to the corresponding state, calculated from DMFT [see Eqs. (37)–(40)] (network simulations). Blue line and blue circles: overlap with a recent memory that is retrieved instead of the memory of age $s$. The DMFT blue line corresponds to Eqs. (37)–(40) for $s=0$. Red line: overlap with a memory of age $s$ in the unstable retrieval state [see Eqs. (22), (27), and (28)]. Vertical dashed gray line: transition to chaos [see Eqs. (22)–(24)]. Vertical dashed blue line: age at which memories become unstable. In simulations, averages are over time and over 10 realizations. (b) Autocovariance parameters versus memory age. Green line (green circles): ${\mathrm{\Delta }}_{0s}={\mathrm{\Delta }}_{1s}$ computed from DMFT (simulations). Blue line (blue circles): ${\mathrm{\Delta }}_{00}={\mathrm{\Delta }}_{10}$ computed from DMFT (simulations). Black line and red line: ${\mathrm{\Delta }}_{0s}={\mathrm{\Delta }}_{1s}$ in unstable retrieval state (DMFT) (c) Dynamics of overlaps for a memory of age $s=5/K=0.18$. The memory is retrieved successfully. (d) Dynamics of overlaps for a memory of age $s=8/K=0.29$. The memory is not retrieved, as the network goes instead to the attractor state corresponding to the most recent memory. In both (c) and (d), overlaps are color coded by age. Network parameters: $N={10}^6$, $K=2\log (N)$, $\tau =0.64$, and $A=4$.

Figure 6

(a) Overlap versus memory age ($s$) for different values of the forgetting timescale $\tau$, and $A=4$. Full lines: overlap computed using DMFT in stable retrieval states [see Eqs. (37)–(40)]. Dashed lines: unstable retrieval states [see Eqs. (22), (27), and (28)]. Circles: simulations (average over time and over 10 realizations). (b) Bifurcation diagram for the network with forgetting, for $A=4$. Black line: maximal age at which retrieval states exist, calculated using DMFT [see Eqs. (29) and (30)]. Blue line (blue circles): transition to chaos of retrieval states [DMFT [see Eqs. (22)–(24)] (simulations)]. Red line (red circles): maximal age at which retrieval states are stable [DMFT [see Eqs. (39)–(41)] (simulations)]. Simulation parameters: $N={10}^6$, $K=2\log (N)$.

Figure 7

Dynamics and robustness of a network with transfer function and learning rules inferred from in vivo data, with forgetting, and external noise. Dynamics and robustness of a network with transfer functions and learning rules inferred from in vivo data, with forgetting, and with external noise. (a) Overlap versus memory age for different values of the standard deviation of external noise ${\sigma }_z$ [see Eq. (43)]. The overlap is calculated by averaging over 50 network realizations for each value of ${\sigma }_z$. Error bars: standard error of the mean. The gray vertical dashed line indicates the transition to chaos for ${\sigma }_z=0$ estimated from numerical simulations. (b) Dynamics of 10 randomly chosen neurons (top row) and of the overlaps with 200 most recent patterns (bottom row) for three different values of the standard deviation of the noise ${\sigma }_z=0$ (red), ${\sigma }_z=1.2$ (green), and ${\sigma }_z=2$ (blue). Each column corresponds to the retrieval of a memory of age indicated at the top of the panel. The same network connectivity is used in (b) for all network simulations. Note that for $s=0.23$, the noise destabilizes the corresponding retrieval state, and the network goes to an attractor corresponding to a more recent memory. Parameters: $N=150000$, $c=1/\sqrt{N}$.

Figure 8

Schematics of the attractor networks phase space. (a) Extensive transition to chaos of fixed-point attractor memory states in an attractor network with no forgetting. When the memory load $\alpha$ surpasses a critical point all the fixed-point attractors transition to chaos at once. For a given value of the memory load $\alpha$, all fixed-point or chaotic memory states are statistically equivalent. (b) In an attractor network that forgets, memory states are heterogeneous. Most recent memories are retrieved as fixed-point attractors with a larger basin of attraction, while older memories are chaotic with smaller basins of attraction. The chaotic temporal fluctuations become faster for older memories. (c) As predicted by our stability analysis, older memory states become unstable after a critical memory age, and the network jumps to the most recent memory state (see the light blue arrows).