2026-09-29

Engram cells in hippocampal CA1 represent contextual information associated with events experienced by animals and constitute a cellular substrate of episodic memory. Recent experiments have shown that a subset of hippocampal place cells is recruited as engram cells for individual contexts. However, the principle governing engram cell allocation across contexts remains unclear. Here, we develop an information-theoretic framework that quantifies the trade-off between spatial and contextual information encoded by CA1 neurons. Our theory shows that limited neural resources inevitably create competition among contexts for the recruitment of place cells as engram cells. Solving this optimization problem determines the fraction of engram cells allocated to each context as a function of its occurrence probability. The theory predicts a non-monotonic relationship between context probability and engram allocation, reflecting competition between spatial and contextual information under resource constraints. Consequently, the recruited fraction emerges through a discontinuous transition at a critical probability, reaches a maximum at intermediate probabilities, and decreases for highly frequent contexts. Furthermore, we analytically derive the phase boundary for the emergence of finite engram cell allocation as a function of contextual input strength and the relative importance of contextual information. These findings identify competition for limited neural resources as a key determinant of engram cell allocation and provide experimentally testable predictions for how memory representations are distributed across contexts under resource constraints.

Traveling waves of activity in neural tissue can be halted by spatial inhomogeneity in synaptic coupling. We study an integrate-and-fire network in which each neuron fires once and the coupling decays exponentially. For this model the leading-edge firing map reduces exactly to a scalar equation for the wave speed in space, with a slow unstable and a fast stable homogeneous speed, $c_1$ and $c_2$, and a periodic modulation of the coupling enters this equation pointwise. Positive periodic waves terminate in folds. For slowly varying modulation the fold amplitude approaches a plateau set by the trough of the modulation, where the local bottleneck speed is $\sqrt{c_1 c_2}$; the approach to the plateau is set by the local geometry of the trough. For rapidly varying modulation the fold amplitude grows linearly with frequency, with a slope given by a full-amplitude average over the integrated modulation profile; a weak-ripple truncation of that average overestimates the slope about 2.5-fold at the default coupling. A perturbative recursion in the amplitude gives the speed profile explicitly. We distinguish loss of the periodic wave from failure of a particular launch, and the excitatory regime $\epsilon\le 1$ from its sign-changing extension. The results are checked against direct integration of the reduced equation and first-spike firing-map simulations of the network.