--- title: 'Excitatory–Inhibitory TLNs: Graphs & Fixed Points' url: https://www.emergentmind.com/topics/excitatory-inhibitory-threshold-linear-networks-e-i-tlns type: topic --- # Excitatory–Inhibitory TLNs: Graphs & Fixed Points 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” [2510.05098], 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” [2606.00373], 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 [2510.05098]. The graph \(G\) determines only the excitatory-to-excitatory connectivity pattern. If \(j\to i\) in \(G\), then excitatory neuron \(j\) excites neuron \(i\); if \(j\not\to i\), there is no direct excitatory coupling. The dynamics are threshold-linear. In the parameterization given in [2510.05098], \[ \frac{dx_i}{dt} = -x_i + \left[ \sum_{j=1}^n W_{ij}x_j + W_{iI}x_I + b_i \right]_+, \qquad i=1,\dots,n, \] \[ \tau_I \frac{dx_I}{dt} = -x_I + \left[ \sum_{j=1}^n W_{Ij}x_j + b_I \right]_+, \] with \([z]_+ = \max\{z,0\}\). The excitatory timescale is normalized to \(1\), while the inhibitory timescale is \(\tau_I\), typically assumed smaller [2510.05098]. The excitatory-to-excitatory weights are defined by \[ W_{ij} = \begin{cases} a_j, & \text{if } j\to i \text{ in } G,\\ 0, & \text{if } j\not\to i \text{ in } G, \end{cases} \qquad i,j\in[n], \] where \(a_j>0\) depends only on the presynaptic excitatory node \(j\). The inhibition-related couplings are \[ W_{Ij}=c_j,\qquad W_{iI}=-1,\qquad W_{II}=0, \] so each excitatory neuron excites the inhibitory node with strength \(c_j\), and the inhibitory node inhibits every excitatory neuron with weight \(-1\) [2510.05098]. A distinctive feature is the excitatory self-coupling \[ W_{ii}=c_i. \] These self-excitation terms are included to cancel the self-inhibition that would otherwise arise through the inhibitory loop. Equivalently, \[ W_{ii}=-W_{iI}W_{Ii}=c_i. \] Unless otherwise specified, the external drives are \[ b_i=\theta>0 \quad (i=1,\dots,n),\qquad b_I=0. \] The admissible parameter constraints are \[ a_j>0,\qquad 1a+1\), moderate inhibition \(10\), \(\delta_j>0\), \(\varepsilon_j<1\), and \(b_i=\theta>0\) for all \(i\) [2510.05098]. The parameter correspondence between the two models is explicit: \[ \varepsilon_j=1+a_j-c_j,\qquad \delta_j=c_j-1, \] and conversely \[ a_j=\varepsilon_j+\delta_j,\qquad c_j=1+\delta_j. \] The E-I conditions \(a_j>0\) and \(10\) and \(\delta_j>0\) in the corresponding gCTLN [2510.05098]. At fixed points, and more generally in the fast-inhibition reduction, the E-I TLN induces an effective \(n\times n\) matrix on excitatory neurons: \[ W_{ij}^{\mathrm{eff}} = W'_{ij}+W'_{iI}W'_{Ij}. \] With the E-I parameterization this becomes \[ W_{ij}=\begin{cases} a_j-c_j,& j\to i,\\ -c_j,& j\not\to i,\\ 0,& i=j, \end{cases} \] which, after substituting \(a_j=\varepsilon_j+\delta_j\) and \(c_j=1+\delta_j\), agrees exactly with the gCTLN weight rule [2510.05098]. Theorem 3 of [2510.05098] states that corresponding gCTLNs and E-I TLNs have the same fixed points in a strong sense. If \(x^*=(x_1^*,\dots,x_n^*)\) is a fixed point of the gCTLN, then \[ \hat x^*=(x_1^*,\dots,x_n^*,x_I^*) \] is the corresponding fixed point of the E-I TLN, where \[ x_I^*=\sum_{j=1}^n W'_{Ij}x_j^*. \] Thus the excitatory coordinates are exactly the same in the two models, and the inhibitory coordinate is uniquely determined by them [2510.05098]. The paper also notes that \(\tau_I\) 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 [2510.05098]. ## 3. Graphical domination, reduction, and irreducible graphs The main graph-theoretic tool transferred to E-I TLNs is graphical domination. For vertices \(j,k\in[n]\), \(k\) graphically dominates \(j\), written \(k>j\), if two conditions hold: for every \(i\neq j,k\), \(i\to j\Rightarrow i\to k\); and \(j\to k\) while \(k\not\to j\) [2510.05098]. 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 \(k\) input dominates \(j\), then no fixed point can have \(x_j^*>0\). It then shows that graphical domination in the underlying graph implies input domination in both gCTLNs and E-I TLNs [2510.05098]. For E-I TLNs, the verification uses the inequalities \[ W_{kj}=a_j > -1+c_j = -1+W_{jj}, \qquad W_{jk}=0 < -1+c_k = -1+W_{kk}, \] together with the common positive drive \(b_k=b_j=\theta\) [2510.05098]. Theorem 1 of [2510.05098] states that if \(j\) is dominated in \(G\), then deleting \(j\) does not change the fixed points: \[ FP(G)=FP(G|_{[n]\setminus j}). \] This is stronger than the statement that \(j\) 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 [2510.05098]. 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 [2510.05098]. 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 \(\widetilde G\) and the corollary \[ FP(G)=FP(\widetilde G). \] 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 \[ x_I^*=\sum_{j=1}^n c_jx_j^*. \] 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 [2510.05098]. 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” [2606.00373] introduces a weak domination rule: if \(k\) weakly dominates \(j\), then \(j\in\sigma\Rightarrow k\in\sigma\) for any fixed-point excitatory support \(\sigma\). 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 [2606.00373]. ## 4. Fixed points, supports, and inhibition regimes For any TLN, a fixed point with support \(\sigma\) is characterized by on-neuron conditions \[ x_i^*=y_i^*>0 \quad (i\in \sigma) \] and off-neuron conditions \[ y_k^*\le 0 \quad (k\notin \sigma), \] where \[ y_i(x)=\sum_{\ell}W_{i\ell}x_\ell+b_i. \] For nondegenerate TLNs, the fixed point with support \(\sigma\) is uniquely determined by \[ x_\sigma^*=(I-W_\sigma)^{-1}b_\sigma \] with off-support coordinates set to zero [2510.05098]. 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 [2510.05098]. The later graph-based analysis in [2606.00373] uses the notation \(\esupp(x^*)\) for the excitatory support. Because \(b_I=0\), \(b_i=\theta>0\), and \(W_{II}=0\), the inhibitory node is active at every fixed point, so the full support is always \(\esupp(x^*)\cup\{I\}\) [2606.00373]. That paper also states that support existence is independent of \(\theta\) and \(\tau_I\); only the graph \(G\) and the parameters \(a,c\) matter for existence, whereas stability depends on \(\tau_I\) [2606.00373]. For the \(n\)-path \(1\to 2\to\cdots\to n\), the fixed-point structure depends sharply on the inhibition regime [2606.00373]. In strong inhibition \(c>a+1\), every nonempty excitatory subset is a support: \[ \FPe(G,a,c)=\{\sigma\subseteq[n]\mid \sigma\neq\emptyset\}. \] In moderate inhibition \(1a+1\), again every nonempty subset is an excitatory support. If \[ \frac{a-1}{n-1}0, \] with corresponding fixed point \[ x^*_{\sigma\cup\{I\}}=\alpha\begin{pmatrix}\mathbf{1}_\sigma\\ |\sigma|c\end{pmatrix}, \qquad \alpha=\frac{\theta}{(|\sigma|-1)c-da+1}. \] This is especially important for the full-support fixed point on cycles, where \(d=1\), so existence reduces to \[ c>\frac{a-1}{n-1} \] [2606.00373]. 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 [2510.05098], fast inhibition \(\tau_I\ll 1\) yields the quasi-steady approximation \[ x_I=\sum_{j=1}^n W_{Ij}x_j, \] since \(W_{Ij}>0\) and \(x_j\ge 0\). Substituting this into the excitatory equations gives an effective \(n\)-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 \(\tau_I\), and that \(\tau_I=0.2\) appears sufficient in the examples shown [2510.05098]. By contrast, when \(\tau_I=1\), E-I TLNs often show synchronized E-I oscillations that obscure graph-structured dynamics [2510.05098]. A later paper turns this observation into a primary object of study. For the singleton E-I TLN, \[ \frac{dx_1}{dt}=-x_1+[cx_1-x_I+\theta]_+, \qquad \tau_I\frac{dx_I}{dt}=-x_I+[cx_1]_+, \] the unique fixed point is \[ (x_1^*,x_I^*)=(\theta,c\theta). \] Its Jacobian is \[ \begin{pmatrix} c-1 & -1\\ c/\tau_I & -1/\tau_I \end{pmatrix}, \] so the fixed point is stable for \[ c<1+\frac{1}{\tau_I} \] and unstable for \[ c>1+\frac{1}{\tau_I}. \] 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 [2606.00373]. This singleton oscillation is treated as the basic E-I oscillation module from which larger network behaviors are built. In strong inhibition \(c>a+1\), singleton fixed points exist for every excitatory node on paths and cycles. When, in addition, \[ c>1+\frac{1}{\tau_I}, \] those singleton fixed points are unstable. The paper argues numerically that this combination is required for sequential chaotic oscillations (SCOs) [2606.00373]. 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 \[ 1\to 2\to \cdots \to n, \] and the sequence terminates at a final attractor. On a cycle, the order is \[ 1\to 2\to \cdots \to n\to 1, \] and the switching continues indefinitely [2606.00373]. 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 \[ z_j=x_{j+1}-x_j,\qquad j=1,\dots,n-1, \] together with the mean variable \[ x_E=\sum_{j=1}^n x_j. \] The \(z\)-mode captures excitatory differences, while the mean mode captures total excitatory activity and its interaction with inhibition [2606.00373]. For cycles, the modes decouple in the full-support chamber and in the all-off excitatory chamber. In the full-support chamber, the \(z\)-mode is stable when \[ c<1-a\cos\!\left(\frac{2\pi}{n}\right), \] and unstable when the inequality reverses. The mean mode is stable when \[ a+c<1+\frac{1}{\tau_I}, \] and unstable otherwise [2606.00373]. This yields a classification of attractors associated with the full-support fixed point: synchronized E-I oscillations when the mean mode destabilizes but the \(z\)-mode remains stable; CTLN-like oscillations when the \(z\)-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 [2606.00373]. 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 [2606.00373]. ## 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” [1812.11581] and “Two ‘correlation games’ for a nonlinear network with Hebbian excitatory neurons and anti-Hebbian inhibitory neurons” [1812.11937] 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, \[ y_\alpha=\sum_i A_{\alpha i}x_i, \] and elimination of inhibition yields effective recurrent competition \[ -(A^\top A)x \] on the excitatory population [1812.11581]. 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 [1812.11937]. Other adjacent work emphasizes structured inhibition, control, or topology. “Selective Inhibition and Recruitment of Linear-Threshold Thalamocortical Networks” [2201.00850] studies controlled linear-threshold networks with saturated activation \([x]_0^m\), explicit inhibitory thalamic routing, and piecewise-affine equilibrium maps. “Emergent organization of receptive fields in networks of excitatory and inhibitory neurons” [2205.13614] 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” [2307.09334] 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 [2205.13614]. 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 \[ \tau T\dot r=-r+k[Wr+h]_+^{.n}, \qquad n>1, \] and studies inhibitory stabilization, determinant conditions, and paradoxical or suppressive effects in one-excitatory/one-inhibitory circuits [1202.6670]. By contrast, “Competition, stability, and functionality in excitatory-inhibitory neural circuits” [2512.05252] explicitly studies asymmetric firing-rate networks with saturated threshold-linear activation \([x]_0^1\), 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 [2512.05252]. Finally, “Classification of 2-node Excitatory-Inhibitory Networks” [2403.02869] 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 \(E\to I\) motif and a recurrent \(E\to I,\ I\dashv E\) feedback pair [2403.02869]. 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 [2606.00373]. 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 [2510.05098]. Source: https://www.emergentmind.com/topics/excitatory-inhibitory-threshold-linear-networks-e-i-tlns