On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs

The elliptic sine-Gordon equation in a half plane

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case. We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation. We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions

Sensory processing and categorization in cortical and deep neural networks

Many recent advances in artificial intelligence (AI) are rooted in visual neuroscience. However, ideas from more complicated paradigms like decision-making are less used. Although automated decision-making systems are ubiquitous (driverless cars, pilot support systems, medical diagnosis algorithms etc.), achieving human-level performance in decision making tasks is still a challenge. At the same time, these tasks that are hard for AI are easy for humans. Thus, understanding human brain dynamics during these decision-making tasks and modeling them using deep neural networks could improve AI performance. Here we modelled some of the complex neural interactions during a sensorimotor decision making task. We investigated how brain dynamics flexibly represented and distinguished between sensory processing and categorization in two sensory domains: motion direction and color. We used two different approaches for understanding neural representations. We compared brain responses to 1) the geometry of a sensory or category domain (domain selectivity) and 2) predictions from deep neural networks (computation selectivity). Both approaches gave us similar results. This confirmed the validity of our analyses. Using the first approach, we found that neural representations changed depending on context. We then trained deep recurrent neural networks to perform the same tasks as the animals. Using the second approach, we found that computations in different brain areas also changed flexibly depending on context. Color computations appeared to rely more on sensory processing, while motion computations more on abstract categories. Overall, our results shed light to the biological basis of categorization and differences in selectivity and computations in different brain areas. They also suggest a way for studying sensory and categorical representations in the brain: compare brain responses to both a behavioral model and a deep neural network and test if they give similar results.National Institute of Mental Health (U.S.) (Grant R37MH087027

Representations of Sensory Signals and Abstract Categories in Brain Networks

Many recent advances in artificial intelligence (AI) are rooted in visual neuroscience. However, ideas from more complicated paradigms like decision-making are less used. Although automated decision-making systems are ubiquitous (driverless cars, pilot support systems, medical diagnosis algorithms etc.), achieving human-level performance in decision making tasks is still a challenge. At the same time, these tasks that are hard for AI are easy for humans. Thus, understanding human brain dynamics during these decision-making tasks and modeling them using deep neural networks could improve AI performance. Here we modelled some of the complex neural interactions during a sensorimotor decision making task. We investigated how brain dynamics flexibly represented and distinguished between sensory processing and categorization in two sensory domains: motion direction and color. We found that neural representations changed depending on context. We also trained deep recurrent neural networks to perform the same tasks as the animals. By comparing brain dynamics with network predictions, we found that computations in different brain areas also changed flexibly depending on context. Color computations appeared to rely more on sensory processing, while motion computations more on abstract categories. Overall, our results shed light to the biological basis of categorization and differences in selectivity and computations in different brain areas.NIMH (Grant R37MH087027