On the Theory of Elasticity of Microinhomogeneous Media with Account for Stochastic Changes in the Connectivity of Constituent Components

The paper proposes a mathematical model aimed at calculating the effective elastic moduli of a micro-inhomogeneous two-component isotropic composite material, which components are connected randomly depending on the level of their relative volumetric contents. A stochastic equation is formulated for the connectivity parameter of the constituent components, according to which, with an increase in the volumetric content of the filler, individual inclusions build the structures of the matrix mixture in the form of interpenetrating frameworks, and then turn into a new binding matrix with individual inclusions from the material of the rest of the old matrix. The algorithm for the numerical solution of this stochastic differential equation is constructed in accordance with the Euler-Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random connectivity function of the constituent components of the composite material. A variant of the method aimed at calculating the mathematical expectation of a random connectivity function of the constituent components has been developed and the corresponding differential equation has been obtained for it. It is shown that the numerical solution of this equation and the average value of the production factor function calculated for all realizations of stochastic trajectories give close numerical values. New macroscopic constitutive relations are found for microinhomogeneous materials with a variable microstructure and their effective elastic moduli are calculated. It is noted that the formulas for these effective elastic moduli generalize the known results for isotropic composite materials. The values of the effective elastic moduli, constructed according to the expressions obtained in the paper, lie within the Khashin-Shtrikman range for the lower and upper bounds of the effective elastic moduli of the composite materials. The numerical analysis of the developed models showed a good agreement with the known experimental data.

Structured connectivity in the cerebellum enables noise-resilient pattern separation

The cerebellum is thought to detect and correct errors between intended and executed commands1-3 and is critical for social behaviors, cognition and emotion4-6. Computations for motor control must be performed quickly to correct errors in real time and should be sensitive to small differences between patterns for fine error correction while being resilient to noise7. Influential theories of cerebellar information processing have largely assumed random network connectivity, which increases the encoding capacity of the network's first layer8-13. However, maximizing encoding capacity reduces resiliency to noise7. To understand how neuronal circuits address this fundamental tradeoff, we mapped the feedforward connectivity in the mouse cerebellar cortex using automated large-scale transmission electron microscopy (EM) and convolutional neural network-based image segmentation. We found that both the input and output layers of the circuit exhibit redundant and selective connectivity motifs, which contrast with prevailing models. Numerical simulations suggest these redundant, non-random connectivity motifs increase discriminability of similar input patterns at a minimal cost to the network's overall encoding capacity. This work reveals how neuronal network structure can balance encoding capacity and redundancy, unveiling principles of biological network architecture with implications for artificial neural network design.

Inhibitory neurons exhibit high controlling ability in the cortical microconnectome

The brain is a network system in which excitatory and inhibitory neurons keep activity balanced in the highly non-random connectivity pattern of the microconnectome. It is well known that the relative percentage of inhibitory neurons is much smaller than excitatory neurons in the cortex. So, in general, how inhibitory neurons can keep the balance with the surrounding excitatory neurons is an important question. There is much accumulated knowledge about this fundamental question. This study quantitatively evaluated the relatively higher functional contribution of inhibitory neurons in terms of not only properties of individual neurons, such as firing rate, but also in terms of topological mechanisms and controlling ability on other excitatory neurons. We combined simultaneous electrical recording (~2.5 hours) of ~1000 neurons in vitro, and quantitative evaluation of neuronal interactions including excitatory-inhibitory categorization. This study accurately defined recording brain anatomical targets, such as brain regions and cortical layers, by inter-referring MRI and immunostaining recordings. The interaction networks enabled us to quantify topological influence of individual neurons, in terms of controlling ability to other neurons. Especially, the result indicated that highly influential inhibitory neurons show higher controlling ability of other neurons than excitatory neurons, and are relatively often distributed in deeper layers of the cortex. Furthermore, the neurons having high controlling ability are more effectively limited in number than central nodes of k-cores, and these neurons also participate in more clustered motifs. In summary, this study suggested that the high controlling ability of inhibitory neurons is a key mechanism to keep balance with a large number of other excitatory neurons beyond simple higher firing rate. Application of the selection method of limited important neurons would be also applicable for the ability to effectively and selectively stimulate E/I imbalanced disease states.

Learning probabilistic neural representations with randomly connected circuits

The brain represents and reasons probabilistically about complex stimuli and motor actions using a noisy, spike-based neural code. A key building block for such neural computations, as well as the basis for supervised and unsupervised learning, is the ability to estimate the surprise or likelihood of incoming high-dimensional neural activity patterns. Despite progress in statistical modeling of neural responses and deep learning, current approaches either do not scale to large neural populations or cannot be implemented using biologically realistic mechanisms. Inspired by the sparse and random connectivity of real neuronal circuits, we present a model for neural codes that accurately estimates the likelihood of individual spiking patterns and has a straightforward, scalable, efficient, learnable, and realistic neural implementation. This model’s performance on simultaneously recorded spiking activity of >100 neurons in the monkey visual and prefrontal cortices is comparable with or better than that of state-of-the-art models. Importantly, the model can be learned using a small number of samples and using a local learning rule that utilizes noise intrinsic to neural circuits. Slower, structural changes in random connectivity, consistent with rewiring and pruning processes, further improve the efficiency and sparseness of the resulting neural representations. Our results merge insights from neuroanatomy, machine learning, and theoretical neuroscience to suggest random sparse connectivity as a key design principle for neuronal computation.

Structured sampling of olfactory input by the fly mushroom body

Associative memory formation and recall in the adult fruit fly Drosophila melanogaster is subserved by the mushroom body (MB). Upon arrival in the MB, sensory information undergoes a profound transformation. Olfactory projection neurons (PNs), the main MB input, exhibit broadly tuned, sustained, and stereotyped responses to odorants; in contrast, their postsynaptic targets in the MB, the Kenyon cells (KCs), are nonstereotyped, narrowly tuned, and only briefly responsive to odorants. Theory and experiment have suggested that this transformation is implemented by random connectivity between KCs and PNs. However, this hypothesis has been challenging to test, given the difficulty of mapping synaptic connections between large numbers of neurons to achieve a unified view of neuronal network structure. Here we used a recent whole-brain electron microscopy (EM) volume of the adult fruit fly to map large numbers of PN- to-KC connections at synaptic resolution. Comparison of the observed connectome to precisely defined null models revealed unexpected network structure, in which a subset of food-responsive PN types converge on individual downstream KCs more frequently than expected. The connectivity bias is consistent with the neurogeometry: axons of the overconvergent PNs tend to arborize near one another in the MB main calyx, making local KC dendrites more likely to receive input from those types. Computational modeling of the observed PN-to-KC network showed that input from the overconvergent PN types is better discriminated than input from other types. These results suggest an ‘associative fovea’ for olfaction, in that the MB is wired to better discriminate more frequently occurring and ethologically relevant combinations of food-related odors.

Impact of functional synapse clusters on neuronal response selectivity

Clustering of functionally similar synapses in dendrites is thought to affect neuronal input-output transformation by triggering local nonlinearities. However, neither the in vivo impact of synaptic clusters on somatic membrane potential (sVm), nor the rules of cluster formation are elucidated. We develop a computational approach to measure the effect of functional synaptic clusters on sVm response of biophysical model CA1 and L2/3 pyramidal neurons to in vivo-like inputs. We demonstrate that small synaptic clusters appearing with random connectivity do not influence sVm. With structured connectivity, ~10–20 synapses/cluster are optimal for clustering-based tuning via state-dependent mechanisms, but larger selectivity is achieved by 2-fold potentiation of the same synapses. We further show that without nonlinear amplification of the effect of random clusters, action potential-based, global plasticity rules cannot generate functional clustering. Our results suggest that clusters likely form via local synaptic interactions, and have to be moderately large to impact sVm responses.

Bilateral Alignment of Receptive Fields in the Olfactory Cortex Points to Non-Random Connectivity

While olfactory sensory neurons expressing the same receptor in the nose converge to the same location in olfactory bulb, projections from the olfactory bulb to the cortex exhibit no recognizable spatial topography. This lack of topography is thought to carry over for interhemispheric connectivity, which originates cortically. If connections to and within the cortex are random, information reaching a cortical neuron from both nostrils will be uncorrelated. Instead, we found that the odor responses of individual neurons to stimulation of both nostrils are highly matched. More surprisingly, odor identity decoding optimized with information arriving from one nostril transfers very well to the other side. Computational analysis shows that such matched odor tuning is incompatible with random connections, but is explained readily by local Hebbian plasticity. Our data reveal that despite the distributed nature of the sensory representation in the olfactory cortex, odor information across the two hemispheres is highly coordinated.

Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time

Networks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical (MH) organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that MH topologies may provide an optimal structure for certain dynamical processes. We revisit a MH network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erdős–Rényi random connectivity structure. We show that this model displays a similar small-world effect as the Kleinberg model, where the connection probability between nodes decays algebraically with distance. We find that there is an optimal structure, in both models, for which the pair-averaged first passage time (FPT) and mean cover time of a discrete-time random walk are minimal, and provide a heuristic explanation for this effect. Finally, we show that analytic predictions for the pair-averaged FPT based on an effective medium approximation fail to reproduce these minima, which implies that their presence is due to a network structure effect.

Spatial and temporal structure of choice representations in primate prefrontal cortex

Divergent accounts of how choices are represented by neural populations have led to conflicting explanations of the underlying mechanisms of decision-making, ranging from persistent, attractor-based dynamics to transient, sequence-based dynamics. To evaluate these mechanisms, we characterize the spatial and temporal structure of choice representations in large neural populations in prefrontal cortex. We find that the pronounced diversity of choice responses across neurons reflects only a few, mostly persistent population patterns recruited at progressively later times before and after a choice. Brief sequential activity occurs during a saccadic choice, but is entirely absent in a delay preceding it. The diversity of choice responses, which could result from almost-random connectivity in the underlying circuits, instead largely reflects the topographical arrangement of response-field properties across the cortical surface. This spatial organization appears to form a fixed scaffold upon which the context-dependent representations of task-specific variables often observed in prefrontal cortex can be learned.