An Attractor-Based Complexity Measurement for Boolean Recurrent Neural Networks

Abstract

We provide a novel refined attractor-based complexity measurement for Boolean recurrent neural networks that represents an assessment of their computational power in terms of the significance of their attractor dynamics. This complexity measurement is achieved by first proving a computational equivalence between Boolean recurrent neural networks and some specific class of -automata, and then translating the most refined classification of -automata to the Boolean neural network context. As a result, a hierarchical classification of Boolean neural networks based on their attractive dynamics is obtained, thus providing a novel refined attractor-based complexity measurement for Boolean recurrent neural networks. These results provide new theoretical insights to the computational and dynamical capabilities of neural networks according to their attractive potentialities. An application of our findings is illustrated by the analysis of the dynamics of a simplified model of the basal ganglia-thalamocortical network simulated by a Boolean recurrent neural network. This example shows the significance of measuring network complexity, and how our results bear new founding elements for the understanding of the complexity of real brain circuits.

The topology of the directed clique complex as a network invariant

“…Activation is indeed a phenomenon happening in a directional way prescribed by the connectivity pattern. The invariant presented should also be considered as a complementary measurement of complexity for the assessment of the computational power of Boolean recurrent neural networks (Cabessa and Villa 2014). …”

Consistency of heterogeneous synchronization patterns in complex weighted networks

“…In the particular case of complex networks of coupled nonlinear oscillators, recent studies have provided evidence that it * Electronic address: daniel.malagarriga@upc.edu; Corresponding author is possible to identify an appropriate interaction regime that allows to collect measured data to infer the underlying network structure based on time-series statistical similarity analysis [11] or connectivity stability analysis [12]. In real-life systems, such as ecological networks [13], brain oscillations [14][15][16][17] or climate interactions [18], various types of complex synchronized dynamics have been observed. Therefore, such a diversity in dynamical relationships between the nodes endows a network with stability, flexibility and robustness against perturbations [19].…”

Expressive Power of Evolving Neural Networks Working on Infinite Input Streams

“…More recently, based on biological as well as theoretical considerations, these studies have been extended to the paradigm of infinite computation [3,[5][6][7][8][9][10]. In this context, the expressive power of the networks is intrinsically related to their attractor dynamics, and is measured by the topological complexity of their underlying neural ω-languages.…”

“…In this context, the expressive power of the networks is intrinsically related to their attractor dynamics, and is measured by the topological complexity of their underlying neural ω-languages. In this case, the Boolean recurrent neural networks provided with certain type specification of their attractors are computationally equivalent to Büchi or Muller automata [8]. The rational-weighted neural nets are equivalent to Muller Turing machines.…”