[Paper Review] The bistable brain: a neuronal model with symbiotic interactions

Ricardo López‐Ruiz, Danièle Fournier-Prunaret|arXiv (Cornell University)|Aug 1, 2012

Nonlinear Dynamics and Pattern Formation3 references3 citations

TL;DR

This paper proposes a minimal neuronal network model using coupled logistic maps with symbiotic (excitatory-inhibitory) coupling to reproduce brain-like bistability—specifically, the sleep-wake cycle. The model exhibits robust, topology- and size-independent bistability between a synchronized active state and a quiescent state, driven by mean-field multiplicative coupling, suggesting a universal mechanism for emergent neural bistability.

ABSTRACT

In general, the behavior of large and complex aggregates of elementary components can not be understood nor extrapolated from the properties of a few components. The brain is a good example of this type of networked systems where some patterns of behavior are observed independently of the topology and of the number of coupled units. Following this insight, we have studied the dynamics of different aggregates of logistic maps according to a particular {\it symbiotic} coupling scheme that imitates the neuronal excitation coupling. All these aggregates show some common dynamical properties, concretely a bistable behavior that is reported here with a certain detail. Thus, the qualitative relationship with neural systems is suggested through a naive model of many of such networked logistic maps whose behavior mimics the waking-sleeping bistability displayed by brain systems. Due to its relevance, some regions of multistability are determined and sketched for all these logistic models.

Motivation & Objective

• To investigate whether bistable dynamics—specifically, sleep-wake-like switching—can emerge in complex networks independently of network size or topology.
• To explore the role of local nonlinear dynamics (logistic maps) and specific coupling schemes in generating global bistability.
• To determine the conditions under which a network transitions between active and inactive synchronized states, modeling neural activation cycles.
• To establish a formal, albeit naive, correspondence between the model's two dynamical states and the brain's waking and sleeping states.
• To analyze the robustness of bistability across different network types, particularly scale-free networks, and identify critical thresholds for switching.

Proposed method

• Model each functional unit (neuron or voxel) as a discrete logistic oscillator with dynamics governed by $ x^{i}_{n+1} = \bar{p}_i x^{i}_n (1 - x^{i}_n) $, representing active or inactive states.
• Implement a local mean-field multiplicative coupling scheme where each node’s dynamics depends on the sum of its neighbors’ activities, with a coupling strength $ \lambda $.
• Use a noise term $ \epsilon $ uniformly distributed over nodes to simulate stochastic perturbations, enabling switching between states.
• Apply the coupling scheme to both small networks (few units) and large-scale networks (e.g., scale-free), analyzing stability and multistability regions.
• Map the system’s behavior across parameter space, identifying three distinct phases: robust, intermediate, and catastrophic, based on the fraction of inactive nodes.
• Use numerical simulations and phase-space analysis to locate basins of attraction and critical thresholds $ \epsilon_c $ and $ \lambda_c $ for switching between states.

Experimental results

Research questions

• RQ1 Can a network of coupled logistic maps exhibit global bistability between active and inactive synchronized states, independent of network size and topology?
• RQ2 What specific coupling mechanism enables the emergence of bistability in a network of nonlinear oscillators with local dynamics?
• RQ3 How does the presence of noise influence the switching between the active and inactive states in the network?
• RQ4 What are the critical thresholds $ \epsilon_c $ and $ \lambda_c $ that trigger a transition from a quiescent to an active state?
• RQ5 Is the bistable behavior robust to changes in network structure, particularly in scale-free networks with heterogeneous connectivity?

Key findings

• The network exhibits global bistability between a synchronized active state and a synchronized inactive state, independent of network size and topology.
• The bistable region is bounded by a robust zone (stable switching), an intermediate zone (sensitive to perturbations), and a catastrophic zone (sudden collapse to inactivity).
• When $ p > 1 $, isolated nodes are self-sustained, preventing global collapse; thus, the catastrophic phase only occurs when $ p \leq 1 $.
• Switching from sleep to wake state occurs when noise amplitude $ \epsilon $ exceeds a critical threshold $ \epsilon_c $, with smaller $ \epsilon_c $ required for higher coupling strength $ \lambda $.
• The transition to the active state resembles a phase transition, with the basin of attraction of the inactive state having a 'hollow cane' structure in phase space.
• The system shows a clear inverse relationship between coupling strength $ \lambda $ and required noise threshold $ \epsilon_c $, indicating that stronger coupling reduces the need for external perturbation to awaken the network.