Sequential chaotic oscillations in excitatory-inhibitory threshold-linear networks

The gist Sequential chaotic oscillations (SCOs) are proposed as a candidate dynamical mechanism for sequential metastability in excitatory-inhibitory threshold-linear networks (E-I TLNs), reflecting a balance between integration and segregation <ref:2606.00373#pg2>.

Sequential Chaotic Oscillations (SCOs)

SCOs arise in E-I TLNs under constant input and consist of a sequence of metastable states whose transition order can be predicted by the underlying graph <ref:2606.00373#pg3>. This behavior is described as sequential chaotic oscillations because it exhibits graph-ordered sequential activity and chaotic dynamics under global E-I oscillations <ref:2606.00373#pg2>. The emergence of SCOs requires unstable singleton fixed points and sufficiently strong inhibition <ref:2606.00373#pg2>.

Fixed Point Structure Analysis

The analysis focuses on characterizing the complete fixed point structure of nondegenerate E-I TLNs on paths and cycles, which provides the dynamical foundation for SCOs <ref:2606.00373#pg2>. The collection of all supports or e-supports is in one-to-one correspondence with the set of all fixed points <ref:2606.00373#pg2>. For E-I TLNs on paths, Theorem 3.1 dictates the fixed point e-supports based on the inhibition regime:

"If c > a + 1(strong inhibition), then FPe(G, a, c) = 2n − 1 and FPe(G, a, c) = ⊆ [n] σ ≠ ∅"

"If 1 < c < a + 1(moderate inhibition), then FPe(G, a, c) = 1 and FPe(G, a, c) = ⊆ ¶[n] "

"If 0 < c < 1(weak inhibition), then FPe(G, a, c) = 1 and FPe(G, a, c) = ⊆ [n] "

E-I TLNs on Cycles

For E-I TLNs on cycles with underlying graph G being an n-cycle (n ≥ 3), Theorem 3.2 describes the fixed point e-supports:

"If c > a + 1(strong inhibition), then FPe(G, a, c) = 2n − 1 and FPe(G, a, c) = ⊆ [n] σ ≠ ∅"

"If a−1/n−1 < c < a + 1, then FPe(G, a, c) = 1 and the unique fixed point support is the full support, FPe(G, a, c) = ⊆ [n] "

If c ≤ (a−1)/n−1, then FPe(G, a, c) = 0

Mode Decomposition and Attractor Classification

To distinguish parameter regimes for attractors associated with the full-support fixed point of E-I TLNs on cycles, a decomposition into the z-mode and mean mode is introduced <ref:2606.00373#pg2>. This decomposition captures excitatory differences (z-mode) and overall network activity (mean mode), allowing classification of attractors including E-I oscillations, CTLN-like oscillations, and flower-like attractors <ref:2606.00373#pg2>. The stability of the full-support fixed point is determined by the stability of both subsystems <ref:2606.00373#pg2>.

Effect of Inhibitory Timescale τI

The inhibitory timescale τI shapes the dynamics, and its limiting cases reveal different behaviors <ref:2606.00373#pg2>. For singleton fixed points in the strong and moderate inhibition regimes, the limit τI → 0 stabilizes all singleton fixed points, whereas the limit τI → ∞ destabilizes them <ref:2606.00373#pg2>. For the full-support fixed point on an n-cycle, fast τI ensures mean mode stability, leading to CTLN-like oscillations or a steady state <ref:2606.00373#pg2>. Slow τI can lead to synchronized E-I oscillations or a flower-like attractor <ref:2606.00373#pg2>.

Chaotic Dynamics and Itinerancy

SCOs are interpreted as a form of chaotic itinerancy, suggesting that the itinerant modules (attractor ruins) may serve as building blocks for complex chaos <ref:2606.00373#pg2>. SCOs exhibit irregular dwell times for metastable states, which is consistent with experimental observations of transient oscillations in neural systems <ref:2606.00373#pg2>. This mechanism suggests that chaotic dynamics does not require high dimensionality and can appear in dimension three <ref:2606.00373#pg2>.

Correspondence to CTLNs

E-I TLNs share structural properties with combinatorial threshold-linear networks (CTLNs), and under the separation-of-timescales assumption τI ≪ τE = 1, an E-I TLN effectively reduces to a CTLN <ref:2606.00373#pg2>. For any E-I TLN in the moderate inhibition regime (1 < c < a + 1), it can be matched to a CTLN with the same fixed-point structure, although stability conditions must be reconsidered due to the dependence on τI <ref:2606.00373#pg2>.

Nondegeneracy and Uniqueness

The analysis relies on nondegeneracy of E-I TLNs, which ensures that each support corresponds to at most one fixed point in a nondegenerate E-I TLN <ref:2606.00373#pg2>. This uniqueness result is extended to E-I TLNs for the entire positive (a, c) parameter plane <ref:2606.00373#pg2>.

Graph Rules and Conditions

The fixed point structure is characterized using new graph rules developed for E-I TLNs, including domination theory and weak domination <ref:2606.00373#pg2>. For example, Lemma 5.4 provides the on-neuron condition for uniform in-degree subgraphs <ref:2606.00373#pg2>. Lemma 5.7 provides sufficient off-neuron conditions for arbitrary subgraphs <ref:2606.00373#pg2>. The analysis excludes the boundary c = a + 1 and c = 1 due to degeneracy issues <ref:2606.00373#pg2>.

Summary of Key Findings

The paper establishes that SCOs occur around unstable singleton fixed points when inhibition is strong, and that for cycles, the full-support fixed point exhibits complex dynamics depending on the relative stability of its z-mode and mean mode <ref:2606.00373#pg2>. The transition between different dynamical states (P1–P7) across parameter regions reveals how E-I oscillations can be transient, irregular, and chaotic <ref:2606.00373#pg2>. The presence of nontrivial graph structure significantly enlarges the parameter region supporting E-I oscillations compared to singleton networks <ref:2606.00373#pg2>. The study provides a minimal framework for interpreting structured manifolds observed in neural population activity through SCOs <ref:2606.00373#pg2>.

How it works

The dynamics of an E-I TLN are governed by the threshold-linear form involving excitatory and inhibitory nodes <ref:2606.00373#pg2>. The system is divided into linear subsystems, Lσ, based on the sign pattern of the terms inside the ReLU activation function <ref:2606.00373#pg2>. The fixed point conditions are characterized by on-neuron and off-neuron conditions derived from these linear subsystems <ref:2606.00373#pg2>.

Fixed Point Characterization

The fixed points are denoted by FP(W, θ, τI) and FPe(W, θ, τI), which are related to the collection of all possible supports or e-supports <ref:2606.00373#pg2>. The support of a fixed point x∗ is defined as the set of active nodes and e-support as the set of active excitatory nodes <ref:2606.00373#pg2>.

Mode Decomposition

The full-support fixed point on an n-cycle decomposes into a z-mode and mean mode, which are analyzed separately in different chambers R[n] and R∅ <ref:2606.00373#pg2>. The stability of the full-support fixed point is stable if and only if both subsystems admit stable fixed points <ref:2606.00373#pg2>.

Stability Criteria

The stability of a singleton fixed point with e-support as a singleton is determined by the reduced Jacobian J, which shows it is stable when c 1 + 1/τI <ref:2606.00373#pg2>.

1. A robust mechanism for modeling sequential metastability can be implemented by designing neural networks using Excitatory-Inhibitory Threshold-Linear Networks (E-I TLNs) on paths or cycles, allowing the system to exhibit sequential chaotic oscillations (SCOs). This enables AI systems to generate structured, graph-ordered sequences of metastable states that are not merely synchronized oscillations.

2. The AI system can dynamically transition between distinct dynamical regimes—such as stable fixed points, E-I oscillations, and chaotic attractors—by adjusting network parameters like the inhibition weight (c) and the excitatory weight (a). This allows for controlled exploration of complex state spaces, moving from simple steady states to high-dimensional chaos.

3. By utilizing the decomposition into z-mode and mean mode, the system can distinguish between excitatory differences and overall network activity, enabling targeted manipulation of specific population dynamics while maintaining control over global network synchronization.

4. The AI can perform sequential information processing by exploiting SCOs, where the transition order determined by the graph dictates the sequence of chaotic states. This provides a mechanism for structured, graph-determined information flow that is robust to initial conditions but exhibits irregular dwell times near metastable states.

5. The system can be trained to recognize and generate different types of emergent attractors—such as flower-like attractors—by tuning the parameters in the full-support chamber, allowing the AI to model complex, non-trivial topological structures in its dynamics.

Metastable states, a phenomenon observed in brain dynamics and many other systems, have been proposed as a key feature of healthy brain function, reflecting a balance between integration and segregation. However, it remains unclear how to capture this behavior within a dynamical-systems framework. In this paper, we propose sequential chaotic oscillations (SCOs), arising in excitatory-inhibitory threshold-linear networks (E-I TLNs), as a candidate dynamical mechanism for sequential metastability. As a simple form of chaotic itinerancy, SCOs occur under constant input and consist of a sequence of metastable states whose transition order can be predicted by the underlying graph. To identify the parameter regime for SCOs, we develop new graph rules for E-I TLNs and use them to characterize the fixed point structure of E-I TLNs on paths and cycles. Our results show that the emergence of SCOs requires unstable singleton fixed points and sufficiently strong inhibition. In addition to SCOs, we find that E-I oscillations need not be synchronized. Motivated by this, we introduce a decomposition into the z-mode and the mean mode, which capture excitatory differences and overall network activity, respectively. These modes are then used to distinguish attractors associated with the full-support fixed point of E-I TLNs on cycles.