Mean Field Theory of Self‐Organizing Memristive Connectomes

Abstract

Biological neuronal networks are characterized by nonlinear interactions and complex connectivity. Given the growing impetus to build neuromorphic computers, understanding physical devices that exhibit structures and functionalities similar to biological neural networks is an important step toward this goal. Self‐organizing circuits of nanodevices are at the forefront of the research in neuromorphic computing, as their behavior mimics synaptic plasticity features of biological neuronal circuits. However, an effective theory to describe their behavior is lacking. This study provides for the first time an effective mean field theory for the emergent voltage‐induced polymorphism of circuits of a nanowire connectome, showing that the behavior of these circuits can be explained by a low‐dimensional dynamical equation. The equation can be derived from the microscopic dynamics of a single memristive junction in analytical form. The effective model is tested on experiments of nanowire networks and show that it fits both the potentiation and depression of these synapse‐mimicking circuits. It is shown that this theory applies beyond the case of nanowire networks by formulating a general mean‐field theory of conductance transitions in self‐organizing memristive connectomes.

Cycle equivalence classes, orthogonal Weingarten calculus, and the mean field theory of memristive systems

“…Although it is well-known that circuits composed of purely memristive devices must also exhibit memristive behavior in two-probe experiments [8], it is less obvious why the collective behavior should resemble that of a single memristive device, especially when the network of devices is disordered. Recent experiments with silver nanowires [25] have confirmed this observation, demonstrating that a mean-field theoretical description of a disordered network of nanowires can accurately describe the potentiation and depression of conductance. On a side note, understanding the phenomenology of memristive networks has direct applications in utilizing these physical systems as reservoir computing devices at the edge of chaos.…”

“…Although this representation might be superfluous, one can always set a voltage generator to zero in the vector⃗ s corresponding to the memristive component's internal memory x i . Although this might seem unphysical at first, a network of nanowires has a very similar representation when the resistance of the nanowire is small compared to the memristive junctions at avoided crossings (see [25], supplementary material) The vector⃗ s represents the voltage generator, while x i are the internal memory parameters of a memristive switch. As a note for the rest of the paper, the bold notation Ω, X distinguishes matrices from scalars or vectors (as for instance the vector of voltages⃗ s).…”

“…This is surprising, considering that these are highly nonlinear systems characterized by conservation laws that give rise to effective nonlocal interactions, albeit exponentially bounded [24]. Such mean-field techniques have also been tested in experimentally viable systems, like nanowire connectomes [25].…”

“…The present manuscript is an attempt to address some of the limitations of previous mean field methods, incorporating the circuit properties into mean field theory. We know, for instance, that in quasi-two-dimensional materials these second-order transitions are only a crossover [25]. One reason for this is that the mean field theoretical results implicitly assume the homogeneity of the network.…”

Ergodicity, lack thereof, and the performance of reservoir computing with memristive networks

“…research has highlighted that nanowire Networks (NWNs) showcase brain-like dynamics, demonstrating their optimal information storage and processing capabilities at conductance transition points [21,23,24]. More recently, a dynamical mean-field theoretical technique for polymer-coated Ag nanowires has uncovered emergent dynamical features [61] such as transitions. In the context of the two-terminal setup, these transitions are commonly observed within a regime termed as the 'edge of formation' [21].…”