P009 Exponential increase of engram cardinality with cell assembly overlap
Jonah G. Ascoli1, Giorgio A. Ascoli2,Rebecca F. Goldin*3
1Lake Braddock Secondary School, Burke, VA USA
2Center for Neural Informatics, George Mason University, Fairfax (VA), USA
3Mathematical Sciences, George Mason University, Fairfax (VA), USA
*Email: rgoldin@gmu.edu
Introduction
Coding by cell assemblies in the nervous system is widely believed to provide considerable computational advantages, including pattern completion and loss resilience [1]. With disjoint cell assemblies, these advantages come at the cost of severely reduced storage capacity relative to single-neuron coding. We prove analytically and demonstrate numerically that allowing a minimal overlap of shared neurons between cell assemblies dramatically boosts network storage capacity.
Methods
Consider a network ofnneurons and an assembly size ofkneurons. Fix a nonnegative numbert<k.Thenetwork capacityCis the engram cardinality: the maximum number of cell assemblies of sizekwith any two assemblies intersecting no more thanttimes.
We find a lower bound forCusing a constructive algorithm. More specifically, we use Lagrange interpolation to construct sets of sizekusing graphs of polynomials over finite fields. The sets have pairwise intersection no larger thantdue to a foundational theorem in algebra. We use standard techniques in combinatorics to determine an upper bound on the network capacity.
Results
We describe the order of magnitude of growth of the network capacity of a system withnneurons, assembly sizekand pairwise overlap of sizet.In the special case that n is equal tok-squared,kis prime, andt=1, we find that the capacity isk(k+1),a(k+1)-fold increase over the easily observable network capacity of k when t=0. We prove more generally that, whent^2 is smaller thank, the network capacity grows liken/kto the powert+1, meaning it is exponential int+1and polynomial inn/k. Without the constraint thattis less than the square root ofk, we show that the network capacity grows liken/kto the powert+1, multiplied byeto the power of an order (t^2/k) function.We design a constructive algorithm that generates sets to actualize the lower bound of the network capacity.
Discussion
Estimates of cell assembly sizes in rodent brains range from~150to~300[2], with larger values in humans. Recent computational work showed that cell assemblies remain representationally distinct when sharing up to 5% of their neurons [5], corresponding tot>7whenk=150. For a network of sizen~20,000, similar to the smallest subregions of the mouse brain [3], we obtain an engram cardinality ~1.7×10^15. With~8distinct mental states per second, corresponding to cortical theta rhythms [4], the engram cardinality is more than 7 orders of magnitude greater than what would suffice to store every single experience in a rodent’s lifetime.
Acknowledgements
This work was supported in part by National Science Foundation (NSF) #2152312 and National Institutes of Health (NIH) R01 NS39600.
References
[1] A. Choucry, M. Nomoto, K. Inokuchi. Engram mechanisms of memory linking and identity.Nature Reviews Neuroscience, 25(6):375-392, Jun 2024.
[2] I. Marco de Almeida, Licurgo, J. E. Lisman. Memory retrieval time and memory capacity of the ca3 network.Learning Memory,2007.
[3] D. Krotov. A new frontier for hopfield networks.Nature Review Physics, 5(7):366–367, Jul 2023.
[4] P. Fries. Rhythmic attentional scanning.Neuron, 111(7):954–970, Apr 2023.
[5] J.D. Kopsick, J.A. Kilgore, G.C. Adam, G.A. Ascoli. Formation and retrieval of cell assemblies. bioRxiv, 2024.
