Related papers: Long cycles in linear thresholding systems
Complex numbers have long been favoured for digital signal processing, yet complex representations rarely appear in deep learning architectures. RNNs, widely used to process time series and sequence information, could greatly benefit from…
A single dynamical system with time-delayed feedback can emulate networks. This property of delay systems made them extremely useful tools for Machine Learning applications. Here we describe several possible setups, which allow emulating…
Many natural systems, such as neurons firing in the brain or basketball teams traversing a court, give rise to time series data with complex, nonlinear dynamics. We can gain insight into these systems by decomposing the data into segments…
Cellular behavior is governed by gene regulatory processes that are intrinsically dynamic and nonlinear, and are subject to non-negligible amounts of random fluctuations. Such conditions are ubiquitous in physical systems, where they have…
In this study, we investigate the continuous time dynamics of Recurrent Neural Networks (RNNs), focusing on systems with nonlinear activation functions. The objective of this work is to identify conditions under which RNNs exhibit perpetual…
Research on long non-coding RNAs (lncRNAs) has garnered significant attention due to their critical roles in gene regulation and disease mechanisms. However, the complexity and diversity of lncRNA sequences, along with the limited knowledge…
Complex systems are characterized by many interacting units that give rise to emergent behavior. A particularly advantageous way to study these systems is through the analysis of the networks that encode the interactions among the system's…
This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.
In this article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…
Can multilayer neural networks -- typically constructed as highly complex structures with many nonlinearly activated neurons across layers -- behave in a non-trivial way that yet simplifies away a major part of their complexities? In this…
We consider evolution of a large population, where fitness of each organism is defined by many phenotypical traits. These traits result from expression of many genes. We propose a new model of gene regulation, where gene expression is…
The vast majority of systems of practical interest are characterised by nonlinear dynamics. This renders the control and optimization of such systems a complex task due to their nonlinear behaviour. Additionally, standard methods such as…
We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor, but in its vicinity as well. For this we consider systems perturbed by an external force. This allows us to not merely…
The inclusion of a threshold in the dynamics of layered neural networks with variable activity is studied at arbitrary temperature. In particular, the effects on the retrieval quality of a self-controlled threshold obtained by forcing the…
Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural…
Regulatory interactions between genes show a large amount of cross-species variability, even when the underlying functions are conserved: There are many ways to achieve the same function. Here we investigate the ability of regulatory…
Threshold-linear networks consist of simple units interacting in the presence of a threshold nonlinearity. Competitive threshold-linear networks have long been known to exhibit multistability, where the activity of the network settles into…
Dynamical systems theory provides a framework for analyzing iterative processes and evolution over time. Within such systems, repetitive transformations can lead to stable configurations, known as attractors, including fixed points and…
A mathematical model for a two-gene regulatory network is derived and several of their properties analyzed. Due to the presence of mixed continuous/discrete dynamics and hysteresis, we employ a hybrid systems model to capture the dynamics…
The paper examines the discrete-time dynamics of neuron models (of excitatory and inhibitory types) with piecewise linear activation functions, which are connected in a network. The properties of a pair of neurons (one excitatory and the…