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Artificial and natural neural network models are a new toolkit which could be potentially have been used for clarifying of complex brain functions. To attend this goal, such models need to be neurobiologically realistic. However, although…

This note is a brief survey of some results of the recent collaboration of neurobiologists and mathematicians dedicated to stimulus reconstruction from neuronal spiking activity. This collaboration, in particular, led to the consideration…

A biologically plausible computational model for color representation is introduced. We present a mechanistic hierarchical model of neurons that not only successfully encodes local hue, but also explicitly reveals how the contributions of…

Inhibitory neurons play a crucial role in maintaining persistent neuronal activity. Although connected extensively through electrical synapses (gap-junctions), these neurons also exhibit interactions through chemical synapses in certain…

Colour vision has long fascinated scientists, who have sought to understand both the physiology of the mechanics of colour vision and the psychophysics of colour perception. We consider representations of colour in anatomically constrained…

Computational models of cortical activity provide insight into the mechanisms of higher-order processing in the human brain including planning, perception and the control of movement. Activity in the cortex is ongoing even in the absence of…

Grid cells in the medial entorhinal cortex (MEC) of the mammalian brain exhibit a strikingly regular hexagonal firing field over space. These cells are learned after birth and are thought to support spatial navigation but also more abstract…

Repeating spatiotemporal spike patterns exist and carry information. Here we investigated how a single spiking neuron can optimally respond to one given pattern (localist coding), or to either one of several patterns (distributed coding,…

The integration of optimization problems within neural network architectures represents a fundamental shift from traditional approaches to handling constraints in deep learning. While it is long known that neural networks can incorporate…

Exactly solvable neural network models with asymmetric weights are rare, and exact solutions are available only in some mean-field approaches. In this article we find exact analytical solutions of an asymmetric spin-glass-like model of…

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable…

Training neural networks involves solving large-scale non-convex optimization problems. This task has long been believed to be extremely difficult, with fear of local minima and other obstacles motivating a variety of schemes to improve…

The backpropagation of error algorithm used to train deep neural networks has been fundamental to the successes of deep learning. However, it requires sequential backward updates and non-local computations, which make it challenging to…

Discrete structures are currently second-class in differentiable programming. Since functions over discrete structures lack overt derivatives, differentiable programs do not differentiate through them and limit where they can be used. For…

Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming…

Neural coding is a key problem in neuroscience, which can promote people's understanding of the mechanism that brain processes information. Among the classical theories of neural coding, the population rate coding has been studied widely in…

The adaptation of neural codes to the statistics of their environment is well captured by efficient coding approaches. Here we solve an inverse problem: characterizing the objective and constraint functions that efficient codes appear to be…

Although artificial neural networks have shown great promise in applications including computer vision and speech recognition, there remains considerable practical and theoretical difficulty in optimizing their parameters. The seemingly…

Connected decision boundaries are useful in several tasks like image segmentation, clustering, alpha-shape or defining a region in nD-space. However, the machine learning literature lacks methods for generating connected decision boundaries…

The high computational complexity and increasing parameter counts of deep neural networks pose significant challenges for deployment in resource-constrained environments, such as edge devices or real-time systems. To address this, we…