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Temporal Complexity of a Hopfield-Type Neural Model in Random and Scale-Free Graphs

Marco Cafiso, Paolo Paradisi

TL;DR

The paper addresses how temporal structure and network topology shape metastable neural dynamics in a bio-inspired Hopfield-type model. It applies Intermittency-Driven Complexity (IDC) and the Event-Driven Diffusion Scaling (EDDiS) framework, combining DFA and Diffusion Entropy analyses to characterize coincidence-event-driven diffusion on scale-free and Erdős-Rényi graphs. The main finding is that random ER graphs can exhibit IDC-like complexity features similar to SF networks, though parameter regimes and noise levels modulate the phenomena; some topology-specific behaviors also emerge (e.g., power-law with cycle in SF). The work advances understanding of how topology influences temporal complexity and has implications for memory storage and learning efficiency in neural-inspired systems and neuromorphic design.

Abstract

The Hopfield network model and its generalizations were introduced as a model of associative, or content-addressable, memory. They were widely investigated both as an unsupervised learning method in artificial intelligence and as a model of biological neural dynamics in computational neuroscience. The complexity features of biological neural networks have attracted the scientific community's interest for the last two decades. More recently, concepts and tools borrowed from complex network theory were applied to artificial neural networks and learning, thus focusing on the topological aspects. However, the temporal structure is also a crucial property displayed by biological neural networks and investigated in the framework of systems displaying complex intermittency. The Intermittency-Driven Complexity (IDC) approach indeed focuses on the metastability of self-organized states, whose signature is a power-decay in the inter-event time distribution or a scaling behaviour in the related event-driven diffusion processes. The investigation of IDC in neural dynamics and its relationship with network topology is still in its early stages. In this work, we present the preliminary results of an IDC analysis carried out on a bio-inspired Hopfield-type neural network comparing two different connectivities, i.e., scale-free vs. random network topology. We found that random networks can trigger complexity features similar to that of scale-free networks, even if with some differences and for different parameter values, in particular for different noise levels

Temporal Complexity of a Hopfield-Type Neural Model in Random and Scale-Free Graphs

TL;DR

The paper addresses how temporal structure and network topology shape metastable neural dynamics in a bio-inspired Hopfield-type model. It applies Intermittency-Driven Complexity (IDC) and the Event-Driven Diffusion Scaling (EDDiS) framework, combining DFA and Diffusion Entropy analyses to characterize coincidence-event-driven diffusion on scale-free and Erdős-Rényi graphs. The main finding is that random ER graphs can exhibit IDC-like complexity features similar to SF networks, though parameter regimes and noise levels modulate the phenomena; some topology-specific behaviors also emerge (e.g., power-law with cycle in SF). The work advances understanding of how topology influences temporal complexity and has implications for memory storage and learning efficiency in neural-inspired systems and neuromorphic design.

Abstract

The Hopfield network model and its generalizations were introduced as a model of associative, or content-addressable, memory. They were widely investigated both as an unsupervised learning method in artificial intelligence and as a model of biological neural dynamics in computational neuroscience. The complexity features of biological neural networks have attracted the scientific community's interest for the last two decades. More recently, concepts and tools borrowed from complex network theory were applied to artificial neural networks and learning, thus focusing on the topological aspects. However, the temporal structure is also a crucial property displayed by biological neural networks and investigated in the framework of systems displaying complex intermittency. The Intermittency-Driven Complexity (IDC) approach indeed focuses on the metastability of self-organized states, whose signature is a power-decay in the inter-event time distribution or a scaling behaviour in the related event-driven diffusion processes. The investigation of IDC in neural dynamics and its relationship with network topology is still in its early stages. In this work, we present the preliminary results of an IDC analysis carried out on a bio-inspired Hopfield-type neural network comparing two different connectivities, i.e., scale-free vs. random network topology. We found that random networks can trigger complexity features similar to that of scale-free networks, even if with some differences and for different parameter values, in particular for different noise levels
Paper Structure (11 sections, 7 equations, 5 figures, 1 table)

This paper contains 11 sections, 7 equations, 5 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. MODEL DESCRIPTION
  3. Network topology: scale-free vs. Erd$\ddot o$s-Rényi
  4. The Grinstein Hopfield-type network model
  5. EVENT-DRIVEN DIFFUSION SCALING ANALYSIS
  6. Neural coincidence events
  7. Detrended Fluctuation Analysis (DFA)
  8. Diffusion Entropy
  9. NUMERICAL SIMULATIONS AND RESULTS
  10. DISCUSSION
  11. CONCLUDING REMARKS

Figures (5)

  • Figure 1: Results of parameter analysis derived from the behaviour of the total activity distribution in ER networks.
  • Figure 2: Results of parameter analysis derived from the behaviour of the total activity distribution in SF network.
  • Figure 3: DFA and DE analyses. Panels (1) and (2) same parameter set $k_0$ = 5, $t_{ref}$ = 10, $b$ = 2, $J$ = 3 and $p_{endo}$ = 0.01 (same as panels (9) of Fig. \ref{['ER_average_activity_total_activity']} and (9) of Fig. \ref{['SF_average_activity_total_activity']}). Panels (3) and (4) same as before but with different noise level $p_{endo} = 0.001$ (same as panels (10) of Fig. \ref{['ER_average_activity_total_activity']} and (8) of Fig. \ref{['SF_average_activity_total_activity']}).
  • Figure 4: (a) Average Activity plots over time and (b) Histograms of Total Activity for all the qualitative behaviours found in ER networks. Parameters for each panel: (1) $k_0$ = 1, $t_{ref}$ = 4, $b$ = 2, $p_{endo}$ = 0.01, and $J$ = 1; (2) $k_0$ = 1, $t_{ref}$ = 4, $b$ = 2, $p_{endo}$ = 0.1, and $J$ = 1; (3) $k_0$ = 2, $t_{ref}$ = 0, $b$ = 2, $p_{endo}$ = 0.01, and $J$ = 1; (4) $k_0$ = 4, $t_{ref}$ = 0, $b$ = 2, $p_{endo}$ = 0.01, and $J$ = 1; (5) $k_0$ = 1, $t_{ref}$ = 4, $b$ = 2, $p_{endo}$ = 0.001, and $J$ = 1; (6) $k_0$ = 3, $t_{ref}$ = 6, $b$ = 3, $p_{endo}$ = 0.1, and $J$ = 2; (7) $k_0$ = 5, $t_{ref}$ = 6, $b$ = 3, $p_{endo}$ = 0.1, and $J$ = 1; (8) $k_0$ = 5, $t_{ref}$ = 6, $b$ = 3, $p_{endo}$ = 0.1, and $J$ = 2; (9) $k_0$ = 5, $t_{ref}$ = 10, $b$ = 2, $p_{endo}$ = 0.01, and $J$ = 3; (10) $k_0$ = 5, $t_{ref}$ = 10, $b$ = 2, $p_{endo}$ = 0.001, and $J$ = 3.
  • Figure 5: (a) Average Activity plots over time and (b) Histograms of Total Activity for all the qualitative behaviours found in SF networks. Parameters for each panel: (1) $k_0$ = 1, $t_{ref}$ = 0, $b$ = 2, $p_{endo}$ = 0.01, and $J$ = 1; (2) $k_0$ = 1, $t_{ref}$ = 0, $b$ = 2, $p_{endo}$ = 0.1, and $J$ = 1; (3) $k_0$ = 5, $t_{ref}$ = 0, $b$ = 3, $p_{endo}$ = 0.01, and $J$ = 2; (4) $k_0$ = 1, $t_{ref}$ = 0, $b$ = 2, $p_{endo}$ = 0.001, and $J$ = 1; (5) $k_0$ = 3, $t_{ref}$ = 0, $b$ = 3, $p_{endo}$ = 0.1, and $J$ = 1; (6) $k_0$ = 3, $t_{ref}$ = 10, $b$ = 3, $p_{endo}$ = 0.1, and $J$ = 2; (7) $k_0$ = 5, $t_{ref}$ = 10, $b$ = 3, $p_{endo}$ = 0.1, and $J$ = 2; (8) $k_0$ = 5, $t_{ref}$ = 10, $b$ = 2, $p_{endo}$ = 0.001, and $J$ = 3; (9) $k_0$ = 5, $t_{ref}$ = 10, $b$ = 2, $p_{endo}$ = 0.01, and $J$ = 3.