Competition, stability, and functionality in excitatory-inhibitory neural circuits

TL;DR

The paper extends energy-based views of neural dynamics to asymmetric excitatory-inhibitory networks by reinterpreting neurons as agents in a game, each minimizing its own energy. It develops a proximal-pseudo-gradient framework for asymmetric FR dynamics, linking neuron-level energies to a network-wide game and establishing stability via Lyapunov diagonal stability. Through analyses of Wilson-Cowan and EI-lateral-inhibition motifs, the work shows how competitive EI interactions can realize robust contrast enhancement and discrete winner-take-all computations, both in small circuits and hierarchical cortical columns. The study also outlines a pathway toward engineering biologically grounded, dynamically stable neural architectures through mechanism design, parameterization, and layered architectures that amplify subtle environmental differences.

Abstract

Energy-based models have become a central paradigm for understanding computation and stability in both theoretical neuroscience and machine learning. However, the energetic framework typically relies on symmetry in synaptic or weight matrices - a constraint that excludes biologically realistic systems such as excitatory-inhibitory (E-I) networks. When symmetry is relaxed, the classical notion of a global energy landscape fails, leaving the dynamics of asymmetric neural systems conceptually unanchored. In this work, we extend the energetic framework to asymmetric firing rate networks, revealing an underlying game-theoretic structure for the neural dynamics in which each neuron is an agent that seeks to minimize its own energy. In addition, we exploit rigorous stability principles from network theory to study regulation and balancing of neural activity in E-I networks. We combine the novel game-energetic interpretation and the stability results to revisit standard frameworks in theoretical neuroscience, such as the Wilson-Cowan and lateral inhibition models. These insights allow us to study cortical columns of lateral inhibition microcircuits as contrast enhancer - with the ability to selectively sharpen subtle differences in the environment through hierarchical excitation-inhibition interplay. Our results bridge energetic and game-theoretic views of neural computation, offering a pathway toward the systematic engineering of biologically grounded, dynamically stable neural architectures.

Figures (13)

• Figure 1: A qualitative comparison of neural interactions and collective actions in symmetric (left) and asymmetric (right) firing rate networks. (A) Dynamics of a recurrent neural network with firing rates $x$, activation function $\Phi$, synaptic matrix $W$, and external input $u$. Firing-rate models capture the temporal evolution of mean neural activity across interconnected populations. (B) Graphic visualization of neural interaction in symmetric neural networks ($W=W^{\top}$). In symmetric neural networks, every pair of neurons interacts reciprocally: if neuron $i$ influences neuron $j$ with strength $w$, neuron $j$ exerts the same influence back on $i$. Symmetry enforces balance and reciprocity across the network, and allows for the existence of a potential for the dynamics directing the flow of the system. (C) Graphic visualization of neural interaction in asymmetric neural networks ($W\neq W^{\top}$). In asymmetric networks, neurons contribute to the collective activity through their outgoing synapses, but reciprocal connections need not exist—or may differ in strength. Interactions are therefore unbalanced, reflecting the structure of real excitatory–inhibitory circuits. (D) Energy in symmetric firing rate networks. Symmetric interactions define a global energy function that all neurons jointly minimize. Neural activity converges toward a minimum-energy configuration, corresponding to a stable equilibrium of the dynamics. (E) Energies in asymmetric firing rate networks. When symmetry is broken, each neuron minimizes its own energy. Their competition shapes a balance point—a Nash equilibrium—where no neuron can unilaterally improve its outcome. This equilibrium need not correspond to a global minimum but represents the best possible compromise given the asymmetry of interactions.
• Figure 2: Graphic interpretation of the $\operatorname{prox}_{\mathrm{E}_{\text{act}}}$ operator action for any fixed external input $u$. (A) Energy landscape associated to a quadratic interaction cost $\mathrm{E}_{\text{int}}$ and activation cost $\mathrm{E}_{\text{act}}$ for a saturated activation function. In particular, $\mathrm{E}_{\text{act}}$ is equal to zero over the entire domain of linear activation, and equal to $+\infty$ over the entire saturated region. The dynamics are therefore steered inside (or at the boundary) of the linear region. Generally, the interaction cost $\mathrm{E}_{\text{int}}$ defines the global landscape over which the neural network dynamics evolve. The activation cost $\mathrm{E}_{\text{act}}(x)$ penalizes infeasible regions in state space, steering the dynamics towards the feasible region. (B) The $\operatorname{prox}$ operator smoothly alters the basic gradient descent on $\mathrm{E}_{\text{int}}$, pushing the dynamics towards the point in the feasible region that is closest - in the Euclidean metric - to the gradient step $x-\nabla \mathrm{E}_{\text{int}}$.
• Figure 3: Self-excitatory weight $w_{EE}$ drives the emergence of limit cycles through Hopf-like bifurcation. Hopf-like bifurcation observed as the E-I firing-rate network transitions from the antagonistic weak decision regime to the antagonistic indecision regime. Plot shows the limit set for each $w_{EE}$ as it is varied starting in the consensual regime ($w_{EE} = 0.5 < 1$) and capturing the point of bifurcation at $w_{EE} = w_{II} + 2 = 2.5$. Parameters used: $w_{EI} = 4, w_{IE} = 2.3, w_{II} = 0.5, u_E = 1, u_I = -0.2$. Solid blue curves represent stable limit sets, while the dashed blue line represents the locus of unstable equilibria.
• Figure 4: Schematic of the Wilson-Cowan model and complete dynamical and game characterization of the exhibited regimes (A) Schematic of the Wilson-Cowan model, with one excitatory neuron ($E$) and one inhibitory neuron ($I$). Excitatory synapses ($w_{EE}$, $w_{IE}$, light blue) are positive, while inhibitory synapses ($w_{II}$, $w_{EI}$, light orange) are negative. Each neuron receives an external input ($u_E$, $u_I$). (B) Summary table of the model’s dynamic regimes and associated game-theoretic interpretation. As synaptic parameters and inputs vary, the system transitions from a unique globally stable Nash equilibrium (high dissipation), to multiple locally stable equilibria, to a single non-Nash equilibrium with spiral convergence, and finally to self-sustained limit cycles when self-excitation is strong. (C–F) Energy surfaces associated with the excitatory (blue) and inhibitory (orange) neurons and the associated dynamic regimes. (C) Unique Nash equilibrium: Both surfaces are convex, and there exists a single point where the action profiles of both neurons cannot be improved. The dynamics naturally evolve along the energy surfaces toward this joint minimum, producing globally asymptotically stable activity. (D) Multiple Nash equilibria: There exist regions where both energy surfaces are locally minimized, but the excitatory neuron may dominate by trapping the system near its global minimum, even when the inhibitory neuron’s energy is near a local maximum. The network dynamics thus settle at one of the locally stable equilibria. (E) Non-Nash spiral convergence: The excitatory and inhibitory energy surfaces have opposing curvature, creating a concave-convex landscape. The resulting competition steers the dynamics in a spiral trajectory toward a unique interior point in the activation domain. (F) Self-sustained oscillations: In the strong self-excitation regime, the concave-convex energy interaction drives the system into a limit cycle, with the dynamics repeatedly circulating along the curved energy surfaces rather than settling at a fixed point.
• Figure 5: Schematic of a $\mathrm{E}^{2}\mathrm{I}$ excitatory-inhibitory circuit and visualization of the interacting Energies under $\mathcal{LDS}$ constraints. (A) Schematic of a minimal $\mathrm{E}^{2}\mathrm{I}$ circuit composed of two excitatory neurons ($E_{1}$, $E_{2}$) interacting through a shared inhibitory interneuron. Each excitatory neuron receives an external input ($u_{E_{1}}$, $u_{E_{2}}$), which is forwarded to the inhibitory neuron. The interneuron in turn relays equal inhibitory feedback to both excitatory cells, effectively suppressing the activity of the neuron receiving the weaker excitatory drive. (B) Energies associated with the reduced excitatory subsystem obtained by eliminating the inhibitory variable through $x_{I}\equiv \bar{x}_{I}(x_{E_{1}},x_{E_{2}})$, under the condition $u_{E_{1}}>u_{E_{2}}$. Both excitatory neurons are characterized by convex individual Energy landscapes, yet $\mathrm{E}^{E_{2}}$ exhibits a local maximum around $x_{E_{2}}=1$, driving its activity toward the inactive state ($x_{E_{2}}=0$). Conversely, $\mathrm{E}^{E_{1}}$ is minimized at $x_{E_{1}}=1$, promoting activation. The joint dynamics settle at the Nash equilibrium defined by the intersection of these tendencies—$x_{E_{1}}=1$, $x_{E_{2}}=0$—in full agreement with the theoretical predictions for $\mathrm{E}^{2}\mathrm{I}$ circuits under the $\mathcal{LDS}$ condition.

Theorems & Definitions (75)

• Definition 1: Best response curve
• Definition 2: Local best response curve
• Definition 3: Nash equilibrium
• Definition 4: Local Nash equilibrium
• Proposition 1
• Definition 5: Saddle point equilibrium
• Definition 6: Graph
• Definition 7: Adjacent Vertices and Degree
• Definition 8: Simple Graph
• Definition 9: Adjacency matrix of a dynamical system