Transmission Neural Networks: Inhibitory and Excitatory Connections

arXiv:2604.04246 · cs.SI, cs.LG, cs.SY, eess.SY, math.DS · Submitted 2026-04-05 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Transmission Neural Networks".

Jane: The gist This work extends the Transmission Neural Network model to incorporate inhibitory connections and neurotransmitter populations,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: Let's talk about what this paper is actually doing, "Transmission Neural Networks: Inhibitory and Excitatory Connections." The title tells you immediately that they are incorporating two types of connections into the network dynamics.

Jane: It’s taking the standard Transmission Neural Network model and explicitly adding in both inhibitory and excitatory parts to see how that changes the outcome.

Lu: The authors are extending prior work by including these new connection types, which means we're looking at a more realistic way neurons communicate than just simple excitation.

Meng: It’s about making the model more faithful to how real biological networks operate, not just a simplified version of them.

Tom: They show that under certain technical assumptions, this extended network model can be perfectly mapped onto another neural network where each neuron has a continuous state with two dimensions.

Jane: That mapping is key because it allows us to analyze the system using the established tools for those continuous state networks.

Lu: They also incorporate neurotransmitter populations into the modeling, which adds another layer of complexity to how these connections are realized physically.

Tom: So, to recap this part, they're showing that adding inhibition and neurotransmitters allows them to find a way to represent the system with continuous two-dimensional states in a neural network.

Jane: And they also establish stability conditions for the resulting limit network model, which is a big step forward for understanding long-term behavior.

The paper's summary: Tom: Moving into what this paper actually summarizes, it lays out the transmission dynamics with these inhibitory and excitatory connections very clearly in equation (two) <ref:2604.04246#pg1>.

Jane: That equation shows how the state of a neuron at one time step depends on whether it got an input from an excitatory neighbor or an inhibitory neighbor.

Lu: The core idea is that a single effective inhibitory connection suppresses excitation, while a single effective excitatory connection helps activate the neuron when inhibition isn't present.

Meng: That means the dynamics aren't just simple addition; the interaction between types of connections matters for how fast things change in the network.

Tom: And they introduce stochasticity by assuming independence properties for transmissions and states, which leads to a specific conditional probability formula in proposition one <ref:2604.04246#pg1>.

Jane: Proposition one is where they give you the math for how likely you are to transition from one state configuration to another, given the current state of all neurons <ref:2604.04246#pg1>.

Lu: They then simplify things by introducing assumptions like (A3) and (A4), which help make the probability update easier to handle.

Tom: These assumptions lead them to define Shannon information states, s i(k) and o i(k), which measure the uncertainty or information content of the system at each step.

Jane: Those states are defined based on whether a neuron has inhibition or excitation happening at that moment, giving us a way to quantify the influence of those connections.

The paper's improvements: Tom: The paper suggests some specific ways to improve this model, starting with how they handle the probability update when things are simplified using assumptions (A3) and (A4).

Jane: They introduce a new way to update the inhibition probability pi i(k+one), which is equivalent to looking at how the existing inhibition probability changes based on new inputs <ref:2604.04246#pg1>.

Lu: This leads to these information states, s i(k) and o i(k), which are defined in equation (twelve) and (thirteen), giving a quantitative measure of the influence of inhibition and excitation.

Meng: If you're looking at this practically, those information states help us track how much influence a specific inhibitory link has on whether a neuron fires next.

Tom: They then show the evolution of these states is characterized by dynamics in equation (sixteen), which links the next state of those information variables to the current state and connection structure.

Jane: This evolution, s i(k+one) = X j in E k i (, s j(k)), shows how the system moves from one state configuration to another over time <ref:2604.04246#pg1>.

Lu: The real power comes when you look at the limit model with infinite neurotransmitters, where they use a Poisson approximation to establish p(k+one) <ref:2604.04246#pg1>.

Tom: In that limit model, proposition four gives us a formula for the probability of excitation p(k+one) involving j(k) and j(k) <ref:2604.04246#pg3>.

Jane: And they provide a compact representation of this dynamics using p(k+one) = B k E k B k I phi((k), (k)) <ref:2604.04246#pg1>.

Conclusion: Tom: So, to wrap up the whole paper "Transmission Neural Networks: Inhibitory and Excitatory Connections," they've established that we can represent neuron excitation probabilities with a neural network where each neuron has a continuous two-dimensional state vector.

Jane: They also handled the complexity of neurotransmitter populations by applying Poisson approximations to create limit models when the number of neurotransmitters at each link goes to infinity.

Lu: The most concrete result they establish is that sufficient conditions for stability and contraction properties of this limit network model are met under certain assumptions like (A5), (A7), and (A8).

Meng: This means that if we meet those conditions, the system's state will converge toward a stable equilibrium point over time, no matter where you start.

Tom: And they also gave us upper bounds on the states of the system in proposition seven and an asymptotic stability result in proposition eight if the connection matrices are invariant.

Jane: These contraction properties are what we needed to prove that this model is a reliable way to study these complex neural systems long-term.

Lu: It sets up a framework where you can use this model for optimal control solutions under Markov decision processes, which is where we see some really exciting potential for AI applications.

Meng: I'm focused on how this translates practically; understanding the contraction properties helps us design systems that are predictable in complex network scenarios.

Lalam: From an information perspective, using the Shannon information states s i(k) and o i(k) gives us a quantitative measure of how much influence inhibition or excitation is having on a neuron's firing probability.

Tom: It’s a lot of math, but fundamentally they’ve shown that this framework is sound for modeling these complex biological interactions.

Jane: It gives us tools to understand how different types of connections—inhibitory and excitatory—shape the overall activity in a network.

cs.SI, cs.LG, cs.SY, eess.SY, math.DS

Submitted: 2026-04-05

Updated: 2026-10-08

Comments: 8 pages

Project page: https://fbullo.github.io/lnd

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 79/100

The gist: The gist This work extends the Transmission Neural Network model to incorporate inhibitory connections and neurotransmitter populations, establishing equivalent representations in neural networks

Key concepts

Transmission Dynamics
This is the core mathematical equation describing how a neuron's state at the next time step is determined by its current state and the activity of its connected neurons. It specifically incorporates both excitatory (activating) and inhibitory (suppressing) connections, allowing for realistic modeling of neural network behavior.
Neurotransmitter Populations
This extension adds complexity by accounting for different realizations of effective synaptic receptions through neurotransmitter populations. This allows the model to capture variability in how signals are transmitted between neurons, moving beyond simple binary connection states to a more nuanced biological representation.
Limit Model
The limit model is a simplified version of the full system achieved by assuming an infinite number of neurotransmitters at each link. By applying Poisson approximations, this model provides a compact mathematical description of the excitation probabilities, making it easier to analyze long-term network behavior.

Terminology

Summary

The gist This work extends the Transmission Neural Network model to incorporate inhibitory connections and neurotransmitter populations, establishing equivalent representations in neural networks with continuous two-dimensional states and deriving stability conditions for the limit network model (Page 1).

Transmission Dynamics with Inhibitory and Excitatory Connections

The transmission dynamics in a network of neurons are modeled by the equation: Xi(k + 1) = 1 − Yj∈E◦k i(1 − WkijXj (k)) × Yj∈I◦k i(1 − WkijXj (k)) (Page 2). This model extends previous work by including inhibitory connections, where E◦k i denotes the set of incoming neighboring nodes of i with excitatory connection that potentially includes node i at step k, and I◦k i denotes the set of incoming neighboring nodes of i with inhibitory connections that potentially includes node i at step k (Page 2). Remark 1 notes that a single effective inhibitory connection to a neuron suppresses its excitation, whereas in the absence of inhibition, a single effective excitatory connection is sufficient to activate it (Page 2).

The stochastic nature of the dynamics is introduced by assuming independence properties for transmissions and states (A1) and (A2) (Page 3). Under these assumptions, the conditional probability of reaching a state configuration q is given by Pr(X(k + 1) = qX(k)) = Yn i=1 qiρi(k + 1) + (1 − qi)(1 − ρi(k + 1)) (Page 3). Proposition 1 states that under assumptions (A1) and (A2), the transition probability to a state configuration q is given by Pr(X(k + 1) = qX(k) = x) = Yn i=1 qiρi(k + 1) + (1 − qi)(1 − ρi(k + 1)) (Page 3).

Dynamics with State Transformation

To further simplify the model, assumptions (A3), (A4), and the notation for inhibition probability πi(k) are introduced, leading to the probability update in equation (8): pi(k + 1) = 1 − Yj∈E◦k i(1 − w kijpj (k)) × Yj∈I◦k i(1 − w kijpj (k)) (Page 3). This is equivalently given by pi(k + 1) = 1 − πi(k + 1) / πi(k) = 1 − Yj∈E◦k i(1 − w kijpj (k))πi(k + 1) (Page 3).

The probability update in equation (8) is equivalently given by pi(k + 1) = Yn i=1 Pr(Xi(k + 1) = qiX(k)) where the equality is due to the conditional independence of the transmissions assumed in (A2) (Page 3). The resulting Shannon information states si(k) and oi(k) are defined as si(k) ≜ (− log 1 − pi(k)) if πi(k) ∈ (0, 1] else 0, and oi(k) ≜ − log πi(k) (Page 4). The evolution of these states is characterized by the dynamics in equation (16): si(k + 1) = Xj∈E◦k i Ψ(w kij e−oj (k), sj (k)) with initial condition si(0) = − log (1 − pi(0)) for all i ∈ [n] (Page 4).

Models with Neurotransmitter Populations

To account for different realizations of effective receptions, the model is generalized to include neurotransmitter populations using equation (18): Xi(k + 1) = 1 − Yj∈E◦k i a kYijl=1 (1 − Wkij(l)Xj (k)) × Yj∈I◦k i a kYijl=1 (1 − Wkij(l)Xj (k)) where a kij denotes the number of neurotransmitters sent from neuron j to neuron i at step k, and Wkij(l) is a binary variable representing the successful reception of the l-th neurotransmitter at step k from neuron j to neuron i when taking 1 (Page 5).

Under assumptions (A5), (A6), (A7), and (A8), the expected state update is equivalent to pi(k + 1) = 1 − Yj∈E◦k i(1 − w kijpj (k))a kij × Yj∈I◦k i(1 − w kijpj (k))a kij (Page 5). The resulting dynamics for the states si and oi are given by equations (26) and (27): si(k + 1) = Xj∈E◦k i a kijΨ(w kij e−oj (k), sj(k)) with initial condition si(0) = − log (1 − pi(0)) for all i ∈ [n] and oi(0) = 0 for all i ∈ [n], if there is no inhibition before the initial step (Page 5).

Limit Model with Infinite Neurotransmitters

The limit model is established by assuming that the probability of transmission w kij depends on the number of transmissions a kij as follows: w kij = λk ij a kij for all i, j ∈ [n] and k ≥ 0 (Page 6). Applying the Poisson approximation yields pi(k + 1) ≈ 1 − Yj∈E◦k i e−λk ij pj (k) Yj∈I◦k i e−λk ij pj (k) for all i ∈ [n] where λ kij = w kij a kij is the rate for Poisson distribution at time k for the synaptic connection from neuron j to neuron i (Page 6).

Proposition 4 states that under assumptions (A5), (A7), and (A8) and (A9), the limit model for the probability of excitation is given by p(k + 1) = e − o¯j (k) / (1 − e − s¯j (k)) where k ∈ [0, T - 1] and Pr(Xi(k) = 1) = e−o¯j (k) / (1 − e−s¯j (k)) for all i ∈ [n] (Page 6). The compact representation of the dynamics is given by p(k + 1) = Bk E⊙Λ k Bk I⊙Λ k ϕ(¯s(k), o¯(k)) where ϕ(s, o) ≜ [σ(s1, o1), · · ·, σ(sn, on)] with σ(si, oi) ≜ e−oi (1 − e−si) for any si, oi ∈ R (Page 7).

Contraction and Stability Properties

Proposition 6 establishes contraction for the system in equations (38) and (39) if Bk E⊙Λ k Bk I⊙Λ k p < 1, ∀k ≥ 0 (Page 8). This implies that the distance between the state at step k+1 and a fixed point decreases as it approaches the fixed point, specifically s¯(k + 1)o¯(k + 1) − s¯∗(k + 1)o¯∗(k + 1) < s¯(k)o¯(k) − s¯∗(k)o¯∗(k), and this property holds for the initial condition as well (Page 8).

Proposition 7 provides an upper bound for the states of the system in equations (38) and (39), stating that for any step k ≥ 1, s¯i(k) ≤ [ΓE(k, 0)¯s(0)]i and o¯i(k) ≤ [(B k−1 I⊙Mk−1)ΓE(k − 1, 0)s(0)]i (Page 8). Proposition 8 indicates that if the adjacency matrices Bk E = BE, Bk I = BI and Λ k = Λ are invariant with respect to the step k ≥ 0, then the system is asymptotically and exponentially stable with respect to the step k at the origin if max i∈[n] λi(BE ⊙Λ) < 1 (Page 8).

Conclusion

The paper concludes that sufficient conditions for stability and contraction properties of the limit network model have been established (Page 1). The probability of neuron excitations for TransNNs with both inhibitory and excitatory connections under technical assumptions can be equivalently represented by neural networks where each neuron has a two-dimensional continuous state vector and each link has the TLogSigmoid activation function in [1] (Page 1). Furthermore, neurotransmitter populations were considered in an extended model, and Poisson approximations were applied to establish limit models when the number of neurotransmitters at each link are infinite (Page 1).

Improvements for AI systems

  1. The improved system can perform optimal control solutions under Markov decision processes by leveraging a representation of excitation probability derived from the limit model, specifically using Equation (20) or (41). This allows for decision-making in complex neural network systems where the probability of a neuron firing is explicitly modeled based on its current state and connection dynamics.

  2. The system can be designed to understand and predict long-term network behavior by utilizing the contraction properties established in Proposition 6, which guarantees that the system in (38) and (39) is contracting, ensuring that trajectories converge toward a stable equilibrium point regardless of the initial conditions within a specific domain.

  3. The improved system can incorporate detailed synaptic mechanisms into its predictive models by using the limit model derived from neurotransmitter populations, which accounts for the number of neurotransmitters at each link are infinite via Equation (41). This allows for modeling neural activity in scenarios where the chemical reality of synaptic transmission is complex and involves a large number of molecules.

  4. The system can be used to characterize the information content associated with inhibition and excitation by utilizing the Shannon information states defined in Equation (12) and (13), which are related to the Shannon information associated with the absence of inhibition from the neighboring neurons at the previous step k−1. This provides a quantitative measure of how much influence inhibitory or excitatory connections have on a neuron's firing probability.

  5. The system can implement state-dependent, non-linear activation functions for synaptic transmission by using the Tuneable Log-Sigmoid (TLogSigmoid) activation function identified in [1], as shown in Equation (16). This allows the network to learn complex input-output relations that are more expressive than simple linear models.

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