Learning and Organization of Memory for Evolving Patterns

Storing memory for molecular recognition is an efficient strategy for responding to external stimuli. Biological processes use different strategies to store memory. In the olfactory cortex, synaptic connections form when stimulated by an odor and establish an associative distributed memory that can be retrieved upon reexposure to the same odors. In contrast, the immune system encodes specialized memory by diverse receptors that can recognize a multitude of evolving pathogens. Despite the mechanistic differences between memory storage in the olfactory system and the immune system, these processes can still be viewed as different information encoding strategies. Here, we develop analytical and numerical techniques for a generalized Hopfield network to probe the utility of distinct memory strategies against both static and dynamic (evolving) patterns. We find that while classical Hopfield networks with distributed memory can efficiently encode a memory of static patterns, they are inadequate against evolving patterns. To follow an evolving pattern, we show that a Hopfield network should use a higher learning rate, which can in turn distort the energy landscape associated with the stored memory attractors. Specifically, we observe the emergence of narrow connecting paths between memory attractors that lead to misclassification of evolving patterns. We demonstrate that compartmentalized networks with specialized subnetworks are the optimal solutions to memory storage for evolving patterns. We postulate that evolution of pathogens may be the reason for the immune system to be encoded in a focused memory, in contrast to the distributed memory used in the olfactory cortex that interacts with mixtures of static odors. Our approach offers a principled framework to study learning and memory retrieval in out-of-equilibrium dynamical systems.

Popular Summary

Biological systems store memories of molecular interactions to efficiently recognize and respond to stimuli. However, the strategies for encoding a memory vary largely. For example, in the olfactory cortex, an odor can stimulate synaptic connections and establish an associative distributed memory that can be retrieved upon reexposure to the same smell. In contrast, the immune system memory uses diverse receptors that can recognize a multitude of evolving pathogens. Despite the differences between these two systems, these processes can still be viewed as information encoding strategies. Here, we present a theoretical framework with artificial neural networks to characterize optimal memory strategies for both static and evolving patterns.

Our approach is a generalization of the energy-based Hopfield-like neural networks, in which memory is stored as the network’s energy minima. We show that while classical Hopfield networks with distributed memory can efficiently encode and retrieve a memory of static patterns, they consistently misclassify evolving patterns. We demonstrate that compartmentalized networks with specialized subnetworks are the optimal solutions to memory storage for evolving patterns.

The contrast between these memory strategies is reflective of the distinct molecular mechanisms used for memory storage in the immune system and in the olfactory cortex. The memory of odor complexes, which can be assumed as static, is stored in a distributed fashion. On the other hand, the immune system, which encounters evolving pathogens, allocates distinct immune cells to store a memory for different types of pathogens.

Our results suggest that evolution of pathogens may be the reason for the immune system to encode a focused memory, in contrast to the distributed memory used in the olfactory cortex that interacts with mixtures of static odors. Our approach also offers a framework to study learning and memory retrieval in out-of-equilibrium dynamical systems, with broad implications for artificial intelligence and deep learning.

Figure 1

Model of working memory for evolving patterns. (a) The targets of recognition are encoded by binary vectors $\{\sigma \}$ of length $L$. Patterns can evolve over time with a mutation rate $\mu$ denoting the fraction of spin flips in a pattern per network update event. (b) Hebbian learning rule is shown for a network $J$, which is presented a set of $N$ patterns $\{{\sigma }^{\alpha }\}$ (colors) over time. At each step, one pattern ${\sigma }^{\alpha }$ is randomly presented to the network, and the network is updated with learning rate $\lambda$ [Eq. (2)]. (c) The energy landscape for networks with distributed memory with optimal learning rate for static (left) and evolving (right) patterns are shown. The equipotential lines are shown in the bottom 2D plane. The energy minima correspond to memory attractors. For static patterns (left), equilibration in the network’s energy landscape drives a pattern toward its associated memory attractor, resulting in an accurate reconstruction of the pattern. For evolving patterns (right), the heightened optimal learning rate of the network results in the emergence of connecting paths (mountain passes) between the energy minima. The equilibration process can drive a pattern through a mountain pass toward a wrong memory attractor resulting in pattern misclassification. (d) A network with distributed memory (left) is compared to a specialized network with multiple compartments (right). To find an associative memory, a presented pattern ${\sigma }^{\alpha }$ with energy $E(J,{\sigma }^{\alpha })$ in network $J$ equilibrates with inverse temperature ${\beta }_H$ in the network’s energy landscape and falls into an energy attractor ${\sigma }_{\mathrm{att}}^{\alpha }$. Memory retrieval is a two-step process in a compartmentalized network (right): First, the subnetwork $J^i$ is chosen with a probability $P_i\sim \exp [-{\beta }_S{E}^i(J^i,{\sigma }^{\alpha })]$, where ${\beta }_S$ is the inverse temperature for this decision. Second, the pattern equilibrates within the subnetwork and falls into an energy attractor ${\sigma }_{\mathrm{att}}^{\alpha }$.

Figure 2

Reduced performance of Hopfield networks in retrieving memory of evolving patterns. (a) The optimal performance of a network ${\mathcal{Q}}^{*}\equiv \mathcal{Q}({\lambda }^{*})$ [Eq. (3)] is shown as a function of the effective mutation rate ${\mu }_{\mathrm{eff}}=N\mu$. The solid lines show the simulation results for networks encountering different number of patterns (colors). The black dotted line shows the naive expectation for the performance solely based on the evolutionary divergence of the patterns ${\mathcal{Q}}_0\approx 1–2{\mu }_{\mathrm{eff}}$, and the colored dashed lines show the expected performance after accounting for the memory lag $g_{\mathrm{lag}}$, ${\mathcal{Q}}_{\mathrm{lag}}\approx 1–2g_{\mathrm{lag}}{\mu }_{\mathrm{eff}}$; see Fig. S5 in the Supplemental Material [38] for more details. (b) The optimal learning rate ${\lambda }^{*}$ is shown as a function of the effective mutation rate. The solid lines are the numerical estimates, the shaded areas indicate the uncertainty in the estimated optimal learning rate, and dashed lines show the theoretical predictions [Eq. (5)]; see Appendix pp1 and Supplemental Material Fig. S2 [38] for the numerical optimization protocol and the estimation of the uncertainty. (c) The mean energy obtained by simulations of randomly ordered patterns (solid lines) and the analytical approximation [Eq. (4)] for ordered patterns (dotted lines) are shown. Error bars show the standard error from the independent realizations (Appendix pp1). The color code for the number of presented patterns is consistent across panels, and the length of patterns is set to $L=800$.

Figure 3

Compartmentalized memory storage. (a) The optimal performance is shown as a function of the effective mutation rate [similar to Fig. 2] for networks with different number of compartments $C$ (colors) ranging from a network with distributed memory $C=1$ (blue) to a 1-to-1 compartmentalized network $C=N$ (red). (b) The optimal (scaled) learning rate ${\lambda }_c/C$ is shown as a function of the effective mutation rate for networks with different number of compartments [colors according to (a)]. Full lines show the numerical estimates, and the dashed line is from the analytical approximation ${\lambda }_c^{*}=\sqrt{8{\mu }_c/(N_c-1})\approx C{\lambda }_1^{*}$. The scaled learning rates collapse on the analytical approximation for all networks except for the 1-to-1 compartmentalized network (red), where the maximal learning rate $\lambda \approx 1$ is used, and each compartment is fully updated upon an encounter with a new version of a pattern. The number of presented patterns is set to $N=32$. We keep $L\times C=\text{const}$, with $L=800$ used for the network with $C=1$.

Figure 4

Phases of learning and memory production. Different optimal memory strategies are shown. (a) The heat map shows the optimal memory performance ${\mathcal{Q}}^{*}$ as a function of the (scaled) Hopfield inverse temperature ${\beta }_{H_c}={\beta }_{H}C$ and the inverse temperature associated with compartmentalization ${\beta }_S$ for networks learning and retrieving a memory of static patterns ($\mu =0$); colors are indicated in the color bar. The optimal performance is achieved by using the optimal strategy (i.e., learning rate ${\lambda }_c^{*}$ and the number of compartments $c^{*}$) for networks at each value of ${\beta }_{H_c}$ and ${\beta }_S$. The three phases of accurate, partial, and no memory are indicated. (b) The heat map shows the memory strategies for the optimal number of compartments (colors as in the legend) corresponding to the memory performance shown in (a). We limit the optimization to the possible number of compartments indicated in the legend to keep $N/C$ an integer. The dashed region corresponds to the case where all strategies perform equally well. Regions of distributed memory ($C=1$) and the 1-to-1 specialized memory ($C=N$) are indicated. The top panel shows the optimal performance ${\mathcal{Q}}^{*}$ of different strategies as a function of the Hopfield inverse temperature ${\beta }_{H_c}$. The right panel shows the mutual information $\mathrm{MI}(\mathrm{\Sigma },\mathcal{C})$ between the set of pattern classes $\mathrm{\Sigma }\equiv \{{\sigma }^{\alpha }\}$ and the set of all compartments $\mathcal{C}$ normalized by the entropy of the compartments $H(\mathcal{C})$ as a function of the inverse temperature ${\beta }_S$; see Appendix pp1. This normalized mutual information quantifies the ability of the system to assign a pattern to the correct compartment. (c),(d) Similar to (a),(b) but for evolving patterns with the effective mutation rate ${\mu }_{\mathrm{eff}}=0.01$. The number of presented patterns is set to $N=32$ (all panels). Similar to Fig. 3, we keep $L\times C=\text{const}$, with $L=800$ used for networks with $C=1$.