Interacting Branching Neural Dynamics

• Interacting Branching Model is a framework that combines probabilistic branching with excitatory/inhibitory neural dynamics to capture complex neural behavior.
• It utilizes continuous-time and discrete models to reveal dynamic regimes such as asynchronous, saturated, and bistable states with bifurcations.
• The model explains observed cortical phenomena including avalanches, criticality, and chaotic transitions through analytical and simulation approaches.

The interacting branching model of neural network dynamics integrates probabilistic branching processes with excitable and inhibitory neural elements, providing a mathematically tractable framework for describing the emergence of rich, nontrivial dynamical regimes observed in neural circuits. These models capture criticality, collective excitability, asynchronous states, bifurcations, and even chaos, reflecting a diversity of behaviors in both large-scale cortical assemblies and small finite-size networks. The framework unifies concepts from stochastic processes, nonlinear dynamics, and network theory to explain fundamental phenomena such as avalanches, rate-propagation, asymmetry-induced multistability, and seizure-like transitions.

1. Core Models and Mathematical Formulation

Branching models generalize the classical Galton–Watson process by embedding nodes with binary (or multi-state) activity on a network with defined excitatory and inhibitory structure. In its simplest form, the continuous-time excitatory–inhibitory branching process (EI-CP) divides $N$ nodes into a fraction $\alpha$ excitatory ($E$) and $1-\alpha$ inhibitory ($I$), evolving according to:

$$\dot \rho_e = -\rho_e + (\alpha - \rho_e)\,f(\lambda(\rho_e - r \rho_i))$$

$$\dot \rho_i = -\rho_i + (1 - \alpha - \rho_i)\,f(\lambda\rho_e)$$

where $f(x) = \max(0, x)$, $\lambda$ is the branching (E→E) strength, and $r$ is the relative I strength on E targets. Extensions to discrete time ($z \in \{0, 1, ..., \tau_r\}$), finite refractory periods, and feedforward network architectures (layered McCulloch–Pitts, mean-field Markov chains) enrich the phase space and permit analytical treatment of high-order modes and finite-size effects (López et al., 2022, Gajic et al., 2012, Williams-García et al., 2022, Fasoli et al., 2015, Goetz et al., 26 Dec 2025).

Microscopic-level descriptions account for coalescence—multiple active parents targeting a single node—which introduces nonlinear scaling in the expected population activity and biases in macroscopic parameter inference (Zierenberg et al., 2019).

2. Dynamical Phase Structure and Bifurcations

Mean-field analysis reveals a rich bifurcation structure:

• Phase boundaries: Transcritical (continuous), saddle-node (discontinuous), and Hopf bifurcations define transitions between quiescent (Q), excitable-quiescent (EQ), asynchronous-active (AS), and saturated-active (A) regimes.
• Codimension-2 points: The intersection of bifurcation curves (e.g., Bogdanov–Takens at $r_t = \sqrt{5}-2$) organizes the phase diagram and creates windows for multi-attractor coexistence.
• Finite-size symmetry-breaking: In small networks, subpopulations (e.g., inhibitory) spontaneously break symmetry via pitchfork bifurcations, leading to multiple, coexisting non-symmetric equilibria and unique routes to multistability and chaos which are absent in the infinite-size (mean-field) limit (Fasoli et al., 2015).

3. Emergence of Cortical Phenomena

Interacting branching models quantitatively parallel empirical cortical phenomena:

• Avalanches: Activity excursions following small perturbations at Q yield avalanche size and duration distributions; AS regimes reproduce experimentally observed "tilted" (skewed) avalanche shapes due to asymmetric eigenvalues of the linearized dynamics (López et al., 2022).
• Excitability and susceptibility: Non-normal Jacobians (as measured by the Henrici index) at low-activity fixed points permit transient amplification ("collective excitability") from finite perturbations.
• Hysteresis and memory: Bistable regions yield hysteresis loops upon adiabatic parameter sweeps, consistent with observed UP-DOWN state transitions and persistent memory traces.
• Partial synchronization: Fluctuation-dominated AS states exhibit partial order as quantified by the Kuramoto-type parameter, matching observations of broad, 1/f$\!^\alpha$-like power spectra in neural data.

4. Criticality, Quasi-criticality, and Universality

A key theoretical insight is the emergence of a critical or quasi-critical regime:

• Criticality: There exists a connectivity threshold—branching parameter $\lambda_c$ or $\kappa_c$—at which the order parameter (activity density) transitions between disorder and persistent activity, with scaling exponents ($\beta = \gamma = 1$) in the mean-field directed percolation class (Goetz et al., 26 Dec 2025).
• Quasi-criticality and information metrics: In presence of nonzero drive, maximal dynamical susceptibility and mutual information peak along a "Widom line" $\kappa^*(p_s)$, even when strict criticality is unattainable, supporting the quasi-criticality hypothesis for adaptive neural coding (Goetz et al., 26 Dec 2025).
• Spectral structure: The approach of the second-largest eigenvalue $\lambda_1$ of the transition matrix to unity flags broad, long-lived modes and is the spectral signature of the "critical" regime in feedforward and recurrent networks (Gajic et al., 2012).

5. Role of Inhibition, Noise, and Biological Realism

Incorporation of inhibition, noise, and network heterogeneity profoundly modifies the dynamical landscape:

• Inhibition: Augments stability, broadens the critical window, and shifts phase boundaries. Diverse inhibitory motifs (e.g., I→E, E→I, hyperarcs) suppress pathological excitatory runaway and facilitate transitions into quasi-criticality. Shortened inhibitory refractory periods lower excitatory output and bolster stability (Goetz et al., 26 Dec 2025, López et al., 2022).
• Noise: Gaussian (local) noise aligns critical parameters, maintains information transfer in high-threshold regimes, and enables stochastic resonance (Gajic et al., 2012).
• Developmental structure and branching: Interstitial axon branching models, incorporating chemotactic guidance cues and activity-dependent rules, recover genetic, topological, and degree distribution features seen in developing networks (Suleymanov et al., 2013).

6. Chaotic Dynamics and Pathological States

Certain parameter regimes support genuinely chaotic attractors, characterized by positive Lyapunov exponents, period-doubling cascades, and fractal basins of attraction:

• Chaotic transition: In the mean-field CBM, chaos emerges for supercritical branching ($\kappa \gtrsim 3.674$ for $\tau_r=2$), with direct calculation of the characteristic Lyapunov exponent (Williams-García et al., 2022, Goetz et al., 26 Dec 2025).
• Universality: The CBM maps, though similar to the Hénon map, belong to a broader class of generalized Hénon-type maps on a compact domain, with state-dependent Jacobians and unique routes to chaos.
• Epilepsy connection: The chaotic and marginally stable regimes coincide with large-amplitude oscillations reminiscent of seizures; periodic modulation of inhibition can synchronize and 'enslave' excitatory populations, suggesting a bifurcation-based mechanism for ictal onset (Goetz et al., 26 Dec 2025).

7. Inference, Coalescence, and Model-Data Interface

From the perspective of data analysis, branching models clarify distinctions between microscopic and macroscopic control parameters:

• Coalescence nonlinearity: Multiple parents activating the same target cause the population-level activity to deviate from linear branching process predictions; this induces bias in estimates of branching parameters.
• Nonlinear inference: Direct nonlinear regression provides unbiased estimation of the microscopic branching parameter, correcting for coalescence-induced bias in conventional linear estimators (Zierenberg et al., 2019).
• Finite-size scaling: Bias in estimators vanishes as $N^{-1/2}$ in absorbing regimes; in driven regimes, the universal non-linear scaling function captures persistent estimator bias even as $N \to \infty$.

• (López et al., 2022) The excitatory-inhibitory branching process: a parsimonious view of cortical asynchronous states, excitability, and criticality
• (Zierenberg et al., 2019) Description of spreading dynamics by microscopic network models and macroscopic branching processes can differ due to coalescence
• (Suleymanov et al., 2013) Modeling of interstitial branching of axonal networks
• (Williams-García et al., 2022) Route to chaos in a branching model of neural network dynamics
• (Gajic et al., 2012) Neutral stability, rate propagation, and critical branching in feedforward networks
• (Goetz et al., 26 Dec 2025) A Minimal Network of Brain Dynamics: Hierarchy of Approximations to Quasi-critical Neural Network Dynamics
• (Fasoli et al., 2015) The complexity of dynamics in small neural circuits