The Witten-Dijkgraaf-Verlinde-Verlinde(WDVV) equations appeared in the study of two-dimensional topological field theoies in the early 1990s. An extension of the WDVV equations, called the open WDVV equations, was introduced by A.Horev and J.P.Solomon (arXiv:1210.4034). In this paper, we give some particular solutions to the open WDVV equations.

Peter Gacs proposed a one-dimensional cellular automaton capable of a robust self-reproduction. Because the automaton is exceptionally large and complicated, very few people have ever succeeded in simulating it on a computer or analyzing its behavior. Here we demonstrate a partial simulation of Gacs' automaton (of Gray's version), discussing its robustness. We also discuss the potential applications of Gacs' framework.

It is well known that bursting activity plays an important role in the processes of transmission of neural signals. In terms of population dynamics, macroscopic bursting can be described using a mean-field approach. Mean field theory provides a useful tool for analysis of collective behavior of a large populations of interacting units, allowing to reduce the description of corresponding dynamics to just a few equations. Recently a new phenomenological model was proposed that describes bursting population activity of a big group of excitatory neurons, taking into account short-term synaptic plasticity and the astrocytic modulation of the synaptic dynamics [1]. The purpose of the present study is to investigate various bifurcation scenarios of the appearance of bursting activity in the phenomenological model. We show that the birth of bursting population pattern can be connected both with the cascade of period doubling bifurcations and further development of chaos according to the Shilnikov scenario, which leads to the appearance of a homoclinic attractor containing a homoclinic loop of a saddle-focus equilibrium with the two-dimensional unstable invariant manifold. We also show that the homoclinic spiral attractors observed in the system under study generate several types of bursting activity with different properties.

The field of information scrambling has seen significant growth over the last decade, where the out-of-time-ordered correlator (OTOC) has emerged as a prominent tool to probe it. In this work, we use bipartite OTOC, a particular form of OTOC, to study information scrambling in the atom-field interaction models and the model of the Ising spin chain interacting with a tilted magnetic field. This is done considering the effects of open quantum systems. A relationship between information scrambling, using bipartite OTOC, and irreversibility, using entropy production, is probed under unitary dynamics. The equivalence of bipartite OTOC with operator entanglement is explicitly shown for the Ising model.

Deep neural networks give us a powerful method to model the training dataset's relationship between input and output. We can regard that as a complex adaptive system consisting of many artificial neurons that work as an adaptive memory as a whole. The network's behavior is training dynamics with a feedback loop from the evaluation of the loss function. We already know the training response can be constant or shows power law-like aging in some ideal situations. However, we still have gaps between those findings and other complex phenomena, like network fragility. To fill the gap, we introduce a very simple network and analyze it. We show the training response consists of some different factors based on training stages, activation functions, or training methods. In addition, we show feature space reduction as an effect of stochastic training dynamics, which can result in network fragility. Finally, we discuss some complex phenomena of deep networks.

Autism Spectrum Disorder (ASD) is characterized by an altered phenotype in social interaction and communication. Additionally, autism typically manifests differently in females as opposed to males: a phenomenon that has likely led to long-term problems in diagnostics of autism in females. These sex-based differences in communicative behavior may originate from differences in neurocomputational properties of brain organization. The present study looked to examine the relationship between one neurocomputational measure of brain organization, the local power-law exponent, in autistic vs. neurotypical, as well as male vs. female participants. To investigate the autistic phenotype in neural organization based on biological sex, we collected continuous resting-state EEG data for 19 autistic young adults (10 F), and 23 controls (14 F), using a 64-channel Net Station EEG acquisition system. The data was analyzed to quantify the 1/f power spectrum. Correlations between power-law exponent and behavioral measures were calculated in a between-group (female vs. male; autistic vs. neurotypical) design. On average, the power-law exponent was significantly greater in the male ASD group than in the female ASD group in fronto-central regions. The differences were more pronounced over the left hemisphere, suggesting neural organization differences in regions responsible for language complexity. These differences provide a potential explanation for behavioral variances in female vs. male autistic young adults.