Excitatory–Inhibitory TLNs: Graphs & Fixed Points

• Excitatory–inhibitory TLNs are recurrent threshold-linear networks where an excitatory graph and a single inhibitory neuron together determine the system's fixed points.
• They are mathematically mapped to generalized CTLNs via explicit parameter correspondences, ensuring that graph-theoretic fixed-point structures are preserved even in noncompetitive settings.
• Depending on inhibition strength and timescale separation, these networks exhibit regimes ranging from stable fixed points to sequential chaotic oscillations driven by systematic graph-guided switching.

Excitatory–inhibitory threshold-linear networks (E-I TLNs) are a graph-based family of recurrent threshold-linear networks in which $n$ excitatory neurons interact through a directed graph $G$, while a single additional inhibitory neuron provides global inhibition to the excitatory population. In the formulation introduced in “On graphical domination for threshold-linear networks with recurrent excitation and global inhibition” (Curto, 6 Oct 2025), E-I TLNs are designed so that their fixed-point structure is determined combinatorially by the excitatory graph $G$, despite the fact that the full network is not competitive. A later development, “Sequential chaotic oscillations in excitatory-inhibitory threshold-linear networks” (Zang et al., 29 May 2026), studies the same model class in parameter regimes that support metastable oscillatory and chaotic dynamics, including graph-ordered switching among attractor ruins.

1. Definition and canonical architecture

An E-I TLN consists of $n$ excitatory nodes indexed by $1,\dots,n$, corresponding to the vertices of a directed graph $G$, together with one inhibitory node $I=n+1$ that does not correspond to a graph vertex and provides global inhibition to all excitatory nodes (Curto, 6 Oct 2025). The graph $G$ determines only the excitatory-to-excitatory connectivity pattern. If $j\to i$ in $G$, then excitatory neuron $G$0 excites neuron $G$1; if $G$2, there is no direct excitatory coupling.

The dynamics are threshold-linear. In the parameterization given in (Curto, 6 Oct 2025),

$G$3

$G$4

with $G$5. The excitatory timescale is normalized to $G$6, while the inhibitory timescale is $G$7, typically assumed smaller (Curto, 6 Oct 2025).

The excitatory-to-excitatory weights are defined by

$G$8

where $G$9 depends only on the presynaptic excitatory node $G$0. The inhibition-related couplings are

$G$1

so each excitatory neuron excites the inhibitory node with strength $G$2, and the inhibitory node inhibits every excitatory neuron with weight $G$3 (Curto, 6 Oct 2025).

A distinctive feature is the excitatory self-coupling

$G$4

These self-excitation terms are included to cancel the self-inhibition that would otherwise arise through the inhibitory loop. Equivalently,

$G$5

Unless otherwise specified, the external drives are

$G$6

The admissible parameter constraints are

$G$7

so an E-I TLN is specified by

$G$8

A major structural contrast with gCTLNs and CTLNs is that the E-I TLN weight matrix is sparse whenever $G$9 is sparse, because missing graph edges correspond to zero excitatory coupling rather than strong inhibition (Curto, 6 Oct 2025).

A later paper studies a graph-based specialization with node-independent parameters $n$0 and $n$1,

$n$2

and homogeneous input $n$3, so that the model is determined by $n$4 (Zang et al., 29 May 2026). That formulation divides parameter space into strong inhibition $n$5, moderate inhibition $n$6, and weak inhibition $n$7 (Zang et al., 29 May 2026).

2. Correspondence with generalized CTLNs

A central result of (Curto, 6 Oct 2025) is that E-I TLNs are tightly linked to generalized combinatorial threshold-linear networks (gCTLNs). A gCTLN is an $n$8-node TLN with weights

$n$9

with $1,\dots,n$0, $1,\dots,n$1, $1,\dots,n$2, and $1,\dots,n$3 for all $1,\dots,n$4 (Curto, 6 Oct 2025).

The parameter correspondence between the two models is explicit: $1,\dots,n$5 and conversely

$1,\dots,n$6

The E-I conditions $1,\dots,n$7 and $1,\dots,n$8 are exactly equivalent to $1,\dots,n$9 and $G$0 in the corresponding gCTLN (Curto, 6 Oct 2025).

At fixed points, and more generally in the fast-inhibition reduction, the E-I TLN induces an effective $G$1 matrix on excitatory neurons: $G$2 With the E-I parameterization this becomes

$G$3

which, after substituting $G$4 and $G$5, agrees exactly with the gCTLN weight rule (Curto, 6 Oct 2025).

Theorem 3 of (Curto, 6 Oct 2025) states that corresponding gCTLNs and E-I TLNs have the same fixed points in a strong sense. If $G$6 is a fixed point of the gCTLN, then

$G$7

is the corresponding fixed point of the E-I TLN, where

$G$8

Thus the excitatory coordinates are exactly the same in the two models, and the inhibitory coordinate is uniquely determined by them (Curto, 6 Oct 2025). The paper also notes that $G$9 does not affect the existence or location of fixed points, though it may affect their stability.

This correspondence is mathematically significant because E-I TLNs are not competitive networks, whereas gCTLNs are. The fixed-point combinatorics nevertheless agree after the parameter mapping. This suggests that the graph-theoretic machinery developed for CTLNs and gCTLNs can be transferred to an explicitly excitatory–inhibitory architecture (Curto, 6 Oct 2025).

3. Graphical domination, reduction, and irreducible graphs

The main graph-theoretic tool transferred to E-I TLNs is graphical domination. For vertices $I=n+1$0, $I=n+1$1 graphically dominates $I=n+1$2, written $I=n+1$3, if two conditions hold: for every $I=n+1$4, $I=n+1$5; and $I=n+1$6 while $I=n+1$7 (Curto, 6 Oct 2025). Intuitively, the dominated node receives no graphically encoded advantage over the dominating node.

The paper first proves a general input-domination criterion for arbitrary TLNs. If $I=n+1$8 input dominates $I=n+1$9, then no fixed point can have $G$0. It then shows that graphical domination in the underlying graph implies input domination in both gCTLNs and E-I TLNs (Curto, 6 Oct 2025). For E-I TLNs, the verification uses the inequalities

$G$1

together with the common positive drive $G$2 (Curto, 6 Oct 2025).

Theorem 1 of (Curto, 6 Oct 2025) states that if $G$3 is dominated in $G$4, then deleting $G$5 does not change the fixed points: $G$6 This is stronger than the statement that $G$7 cannot appear in a fixed point. Every fixed point of the reduced network survives unchanged in the full network, with the deleted node simply remaining off (Curto, 6 Oct 2025).

Theorem 2 states that if dominated nodes are removed iteratively until no dominated nodes remain, the final domination-free reduced graph is unique, independent of the order of removal (Curto, 6 Oct 2025). The proof rests on transitivity of domination and inheritance of domination to induced subgraphs containing the relevant vertices. Consequently, one obtains a uniquely defined reduced graph $G$8 and the corollary

$G$9

For E-I TLNs this yields a practical fixed-point workflow. One reduces the excitatory graph by graphical domination, computes fixed points only on the irreducible graph, and then lifts those fixed points back to the full E-I network by setting deleted excitatory coordinates to zero and reconstructing the inhibitory coordinate via

$j\to i$0

The paper emphasizes that it is not only the supports but also the fixed-point values on surviving excitatory coordinates that are preserved under reduction (Curto, 6 Oct 2025).

A later paper extends the graph-theoretic picture beyond the moderate-inhibition regime. In the moderate regime, the domination rule continues to apply. In the weak-inhibition regime, “Sequential chaotic oscillations in excitatory-inhibitory threshold-linear networks” (Zang et al., 29 May 2026) introduces a weak domination rule: if $j\to i$1 weakly dominates $j\to i$2, then $j\to i$3 for any fixed-point excitatory support $j\to i$4. That paper also gives on-neuron and off-neuron conditions for uniform in-degree subgraphs and disjoint unions of paths, producing complete fixed-point descriptions for paths and cycles (Zang et al., 29 May 2026).

4. Fixed points, supports, and inhibition regimes

For any TLN, a fixed point with support $j\to i$5 is characterized by on-neuron conditions

$j\to i$6

and off-neuron conditions

$j\to i$7

where

$j\to i$8

For nondegenerate TLNs, the fixed point with support $j\to i$9 is uniquely determined by

$G$0

with off-support coordinates set to zero (Curto, 6 Oct 2025). In the E-I setting, one usually tracks supports on the excitatory nodes, since the inhibitory node is auxiliary and determined by the excitatory state (Curto, 6 Oct 2025).

The later graph-based analysis in (Zang et al., 29 May 2026) uses the notation $G$1 for the excitatory support. Because $G$2, $G$3, and $G$4, the inhibitory node is active at every fixed point, so the full support is always $G$5 (Zang et al., 29 May 2026). That paper also states that support existence is independent of $G$6 and $G$7; only the graph $G$8 and the parameters $G$9 matter for existence, whereas stability depends on $G$00 (Zang et al., 29 May 2026).

For the $G$01-path $G$02, the fixed-point structure depends sharply on the inhibition regime (Zang et al., 29 May 2026). In strong inhibition $G$03, every nonempty excitatory subset is a support: $G$04 In moderate inhibition $G$05, the unique excitatory support is $G$06. In weak inhibition $G$07, the unique excitatory support is the full set $G$08 (Zang et al., 29 May 2026).

For the $G$09-cycle $G$10, the picture is different. In strong inhibition $G$11, again every nonempty subset is an excitatory support. If

$G$12

then the unique support is full support $G$13. If

$G$14

there is no fixed point, and the paper reports blow-up of activity (Zang et al., 29 May 2026).

For uniform in-degree subgraphs $G$15 of in-degree $G$16, the paper gives an explicit existence condition: $G$17 with corresponding fixed point

$G$18

This is especially important for the full-support fixed point on cycles, where $G$19, so existence reduces to

$G$20

(Zang et al., 29 May 2026).

These results show that E-I TLNs admit a graph-organized support theory that is not confined to the moderate-inhibition regime corresponding to CTLNs. A plausible implication is that the explicit inhibitory node preserves the graph-theoretic organization of fixed points while exposing new dynamical regimes once inhibition becomes weak or sufficiently strong.

5. Dynamical regimes, oscillations, and sequential chaotic oscillations

The dynamical behavior of E-I TLNs is strongly shaped by timescale separation and inhibition strength. In (Curto, 6 Oct 2025), fast inhibition $G$21 yields the quasi-steady approximation

$G$22

since $G$23 and $G$24. Substituting this into the excitatory equations gives an effective $G$25-dimensional TLN whose matrix is exactly the corresponding gCTLN matrix. The paper reports that E-I TLNs and gCTLNs with the same graph exhibit very similar dynamics, especially for sufficiently small $G$26, and that $G$27 appears sufficient in the examples shown (Curto, 6 Oct 2025). By contrast, when $G$28, E-I TLNs often show synchronized E-I oscillations that obscure graph-structured dynamics (Curto, 6 Oct 2025).

A later paper turns this observation into a primary object of study. For the singleton E-I TLN,

$G$29

the unique fixed point is

$G$30

Its Jacobian is

$G$31

so the fixed point is stable for

$G$32

and unstable for

$G$33

When it becomes unstable, boundedness plus a Poincaré–Bendixson argument imply the existence of a periodic orbit, and the paper numerically finds a stable E-I limit cycle (Zang et al., 29 May 2026).

This singleton oscillation is treated as the basic E-I oscillation module from which larger network behaviors are built. In strong inhibition $G$34, singleton fixed points exist for every excitatory node on paths and cycles. When, in addition,

$G$35

those singleton fixed points are unstable. The paper argues numerically that this combination is required for sequential chaotic oscillations (SCOs) (Zang et al., 29 May 2026).

SCOs are defined as chaotic oscillatory dynamics under constant input that spend long but finite times near a sequence of metastable states, then switch to the next one in an order predicted by the graph. On a path, the order is

$G$36

and the sequence terminates at a final attractor. On a cycle, the order is

$G$37

and the switching continues indefinitely (Zang et al., 29 May 2026). The metastable states are interpreted as attractor ruins in the sense of chaotic itinerancy.

The same paper emphasizes that E-I oscillations need not be synchronized across excitatory nodes. To separate global E-I oscillations from excitatory pattern formation, it introduces the difference coordinates

$G$38

together with the mean variable

$G$39

The $G$40-mode captures excitatory differences, while the mean mode captures total excitatory activity and its interaction with inhibition (Zang et al., 29 May 2026).

For cycles, the modes decouple in the full-support chamber and in the all-off excitatory chamber. In the full-support chamber, the $G$41-mode is stable when

$G$42

and unstable when the inequality reverses. The mean mode is stable when

$G$43

and unstable otherwise (Zang et al., 29 May 2026). This yields a classification of attractors associated with the full-support fixed point: synchronized E-I oscillations when the mean mode destabilizes but the $G$44-mode remains stable; CTLN-like oscillations when the $G$45-mode destabilizes but the mean mode remains stable; and either synchronized E-I oscillations or flower-like quasi-periodic attractors when both modes are unstable (Zang et al., 29 May 2026).

These results broaden the interpretation of E-I TLNs. They are not only graph-theoretic fixed-point models, but also a minimal piecewise-linear setting for metastability, chaotic itinerancy, synchronized and unsynchronized E-I oscillation, and graph-predictable switching under constant tonic drive (Zang et al., 29 May 2026).

6. Relation to adjacent excitatory–inhibitory thresholded network frameworks

Several related arXiv papers situate E-I TLNs within a broader landscape of excitatory–inhibitory thresholded dynamics. The 2018 papers “Unsupervised learning by a nonlinear network with Hebbian excitatory and anti-Hebbian inhibitory neurons” (Seung, 2018) and “Two ‘correlation games’ for a nonlinear network with Hebbian excitatory neurons and anti-Hebbian inhibitory neurons” (Seung, 2018) describe rectified rate networks with explicit excitatory and inhibitory populations, nonnegative activities, sign-constrained connectivity, and inhibition-mediated competition. In those models the inhibitory variables are instantaneous linear readouts,

$G$46

and elimination of inhibition yields effective recurrent competition

$G$47

on the excitatory population (Seung, 2018). Those papers are not canonical E-I TLN papers, but they are close variants in which thresholded E-activity, Dale-type signs, and disynaptic inhibition are central (Seung, 2018).

Other adjacent work emphasizes structured inhibition, control, or topology. “Selective Inhibition and Recruitment of Linear-Threshold Thalamocortical Networks” (McCreesh et al., 2022) studies controlled linear-threshold networks with saturated activation $G$48, explicit inhibitory thalamic routing, and piecewise-affine equilibrium maps. “Emergent organization of receptive fields in networks of excitatory and inhibitory neurons” (Lufkin et al., 2022) develops a local E/I recurrent sparse-coding network with soft-thresholded inference, short-range excitation, broader inhibition, and piecewise-linear active-set structure. “Approximating nonlinear functions with latent boundaries in low-rank excitatory-inhibitory spiking networks” (Podlaski et al., 2023) moves to a spiking formulation, but derives low-dimensional thresholded rate equations in the soft-boundary limit, with stable inhibitory and unstable excitatory latent boundaries. These works are best viewed as adjacent rather than identical to graph-based E-I TLNs (Lufkin et al., 2022).

The broader rate-model literature also contains closely related E/I frameworks that are not threshold-linear in the strict ReLU sense. The stabilized supralinear network analyzes

$G$49

and studies inhibitory stabilization, determinant conditions, and paradoxical or suppressive effects in one-excitatory/one-inhibitory circuits (Ahmadian et al., 2012). By contrast, “Competition, stability, and functionality in excitatory-inhibitory neural circuits” (Betteti et al., 4 Dec 2025) explicitly studies asymmetric firing-rate networks with saturated threshold-linear activation $G$50, derives P-matrix and Lyapunov diagonal stability criteria, and interprets asymmetric E-I dynamics through neuron-wise energies and game-theoretic equilibria. On any region where upper saturation is inactive, those dynamics reduce exactly to standard threshold-linear dynamics (Betteti et al., 4 Dec 2025).

Finally, “Classification of 2-node Excitatory-Inhibitory Networks” (Aguiar et al., 2024) is not a TLN paper, but it classifies connected 2-node E/I motifs in the coupled-cell framework. For restricted E/I networks, all connected 2-node motifs collapse to two ODE-classes: a feedforward $G$51 motif and a recurrent $G$52 feedback pair (Aguiar et al., 2024). That result is structurally relevant because the recurrent feedback pair is the minimal E-I motif already visible in the singleton E-I TLN and in the local oscillation module used in later E-I TLN dynamics (Zang et al., 29 May 2026).

Taken together, these neighboring literatures show that E-I TLNs occupy a specific position within thresholded E/I network theory: they combine an explicit globally inhibitory node, a graph-defined excitatory subnetwork, exact graph-to-fixed-point correspondences, and a piecewise-linear dynamics rich enough to support both combinatorial fixed-point analysis and nontrivial oscillatory and chaotic behavior (Curto, 6 Oct 2025).