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Transmission Neural Networks: Inhibitory and Excitatory Connections

Shuang Gao, Peter E. Caines

Abstract

This paper extends the Transmission Neural Network model proposed by Gao and Caines in [1]-[3] to incorporate inhibitory connections and neurotransmitter populations. The extended network model contains binary neuronal states, transmission dynamics, and inhibitory and excitatory connections. Under technical assumptions, we establish the characterization of the firing probabilities of neurons, and show that such a characterization considering inhibitions can be equivalently represented by a neural network where each neuron has a continuous state of dimension 2. Moreover, we incorporated neurotransmitter populations into the modeling and establish the limit network model when the number of neurotransmitters at all synaptic connections go to infinity. Finally, sufficient conditions for stability and contraction properties of the limit network model are established.

Transmission Neural Networks: Inhibitory and Excitatory Connections

Abstract

This paper extends the Transmission Neural Network model proposed by Gao and Caines in [1]-[3] to incorporate inhibitory connections and neurotransmitter populations. The extended network model contains binary neuronal states, transmission dynamics, and inhibitory and excitatory connections. Under technical assumptions, we establish the characterization of the firing probabilities of neurons, and show that such a characterization considering inhibitions can be equivalently represented by a neural network where each neuron has a continuous state of dimension 2. Moreover, we incorporated neurotransmitter populations into the modeling and establish the limit network model when the number of neurotransmitters at all synaptic connections go to infinity. Finally, sufficient conditions for stability and contraction properties of the limit network model are established.

Paper Structure

This paper contains 10 sections, 8 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Transmission Dynamics with Inhibitory and Excitatory Connections
  3. Transmission Dynamics
  4. Stochasticity in the Transmission Dynamics
  5. Dynamics with State Transformation
  6. Models with Neurotransmitter Populations
  7. Limit Model with Infinite Neurotransmitters
  8. Limit Model via Poisson Approximation
  9. Contraction and Stability Properties
  10. Conclusion

Key Result

proposition 1

Assume (A1) and (A2) hold. Given a state configuration ${x} \in \{0,1\}^n$ at step $k$, the transition probability to a state configuration $q\in\{0,1\}^n$ is given by where and $w_{ij}^k\triangleq \textup{Pr}(W_{ij}^k=1 {| X_j(k)=1})$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure B1: NOR gate with inputs $A$ and $B$, and output $C$, created with inhibitory connections and a constant signal 1.

Theorems & Definitions (28)

  • remark 1
  • remark 2: Functional Completeness
  • remark 3
  • proposition 1
  • proof
  • remark 4
  • remark 5
  • remark 6: Convexity
  • proposition 2
  • remark 7
  • ...and 18 more