Wilson-Cowan with Co-Transmission
Extending the stochastic Wilson-Cowan model to neurons that release both excitatory and inhibitory neurotransmitters
Our Journey: Adding Co-Transmission
Introduction: From 2 Populations to 3
You've learned the Wilson-Cowan model with its E and I populations. The key insight was that E/I balance determines criticality — at w0c = α/β, the brain operates at the “edge of chaos.”
The Big Question
What if neurons could release both excitatory AND inhibitory neurotransmitters?
This is co-transmission — and it's more common in the brain than we once thought. This tutorial extends our Wilson-Cowan foundation to include a third population (C) of co-transmitting neurons.
Wilson-Cowan Tutorial
Two populations:
- E: Excitatory only
- I: Inhibitory only
Dale's Law: neurons have one identity
This Tutorial
Three populations:
- E: Excitatory only
- I: Inhibitory only
- C: Both!
Co-transmission: neurons release both transmitters
What We'll Learn
- Why co-transmission exists biologically
- How to extend Wilson-Cowan from 2 to 3 populations
- The wc_ratio parameter: C's E/I balance
- How C affects the critical point
- Side-by-side comparison of mono vs co-transmission dynamics
Recap: The Stochastic Wilson-Cowan Model
Before adding co-transmission, let's establish the foundation we're building on. The Wilson-Cowan model has two equivalent formulations:
Microscopic vs Macroscopic Formulations
State: Each neuron i has ai ∈ {0, 1}
Transitions:
Active → Quiescent: rate = α
Quiescent → Active: rate = f(s)
Input: s = wE·SE − wI·SI + h
State: S = (1/N) × Σ ai (fraction active)
Langevin Equation:
dS/dt = −αS + (1−S)f(s) + η(t)
Noise term η(t):
η(t) = (1/√N) · √[αS + f(s)(1−S)] · ξ(t)
where ξ(t) is Gaussian white noise with ⟨ξ(t)⟩ = 0 and ⟨ξ(t)ξ(t')⟩ = δ(t−t')
The noise η(t) arises from finite-size fluctuations in the neuron population. It has two key properties:
- Amplitude ∝ 1/√N: Larger populations have smaller relative fluctuations
- Variance ∝ αS + f(s)(1−S): Proportional to the total transition rate (both deactivations and activations)
Key Variables (from Wilson-Cowan Tutorial)
| SE, SI | Fraction of active E and I neurons |
| α | Deactivation rate = 0.1 ms⁻¹ |
| β | Activation gain = 1.0 ms⁻¹ |
| w0 | Net coupling = wE − wI |
| w0c | Critical point = α/β = 0.1 |
| η(t) | Finite-size noise term (scales as 1/√N) |
Section 0 Summary
- Microscopic: Individual neurons with binary states, stochastic transitions
- Macroscopic: Langevin equation for population activity S
- Noise: η(t) = (1/√N) · √[αS + f(s)(1−S)] · ξ(t) from finite-size effects
- Critical point: w0c = α/β = 0.1
- Now we extend this by adding a third population C
Biological Foundation: Co-Transmission
Dale's Principle — The Old View
For decades, neuroscience assumed Dale's Principle: each neuron releases only one type of neurotransmitter.
E Neurons (Glutamate)
- Release glutamate
- Excite postsynaptic targets
- ~80% of cortical neurons
- Excitatory effect (+)
I Neurons (GABA)
- Release GABA
- Inhibit postsynaptic targets
- ~20% of cortical neurons
- Inhibitory effect (−)
The Modern View: Co-Transmission
Recent research shows that many neurons release multiple neurotransmitters. This is co-transmission.
Co-transmission provides fine-grained E/I control at the single-neuron level. Instead of requiring precise tuning of separate E and I populations, a single C neuron can dynamically adjust its E/I balance via wc_ratio.
Section 1 Summary
- Dale's Principle: Neurons are either E or I (old view)
- Co-transmission: Some neurons release BOTH transmitters (new view)
- Examples: VIP+ interneurons, developing neurons, hypothalamic circuits
- Why it matters: Fine-grained E/I control, potential for homeostatic regulation
The Three-Population Model
Standard Wilson-Cowan: Two Populations
Extended Model: Three Populations
New Variables Introduced
| C population | Co-transmitting neurons that release BOTH glutamate and GABA |
| NC | Number of co-transmitting neurons |
| SC | Fraction of active C neurons = m/NC |
| m | Number of currently active C neurons (0 ≤ m ≤ NC) |
| r (wc_ratio) | Fraction of C's output that is excitatory. Range: [0, 1] r = 1.0: Fully excitatory (acts like E) r = 0.5: Half E, half I r = 0.0: Fully inhibitory (acts like I) |
C's New Contribution
Unlike E (purely +) or I (purely −), C contributes BOTH:
Excitatory part: +wE · r · SC
Inhibitory part: −wI · (1 − r) · SC
Net contribution:
wC,net = wE · r − wI · (1 − r)
The Extended Synaptic Input
s = wE·SE − wI·SI + h
s = wE·SE − wI·SI + wC,net·SC + h
The extension from 2-pop to 3-pop is minimal: we simply add the term wC,net·SC to the synaptic input. The C population's net effect depends entirely on wc_ratio (r).
Section 2 Summary
- Standard: s = wE·SE − wI·SI + h
- Extended: s = wE·SE − wI·SI + wC,net·SC + h
- C neurons: Release both glutamate AND GABA
- New variables: NC, SC, m, and wc_ratio (r)
- wC,net = wE·r − wI·(1−r): C's net effect on the network
Gillespie Dynamics with Co-Transmission
The Gillespie algorithm simulates individual neuron transitions. We extend it from 2 to 3 populations.
Active Neuron Counts
| k | Number of active E neurons (0 ≤ k ≤ NE) |
| l | Number of active I neurons (0 ≤ l ≤ NI) |
| m | Number of active C neurons (0 ≤ m ≤ NC) |
Transition Rates
Active → Quiescent:
E: k·α I: l·α
Quiescent → Active:
E: (NE−k)·f(s) I: (NI−l)·f(s)
where s = wE·SE − wI·SI + h
Active → Quiescent:
E: k·α I: l·α C: m·α
Quiescent → Active:
E: (NE−k)·f(s) I: (NI−l)·f(s) C: (NC−m)·f(s)
where s = wE·SE − wI·SI + wC,net·SC + h
Symmetric Input Structure
sE = wE·SE − wI·SI + wC,net·SC + h
sI = wE·SE − wI·SI + wC,net·SC + h
sC = wE·SE − wI·SI + wC,net·SC + h
Note: All populations receive the SAME input (symmetric, like de Candia)
Langevin Equations (3-population)
dSE/dt = −αSE + (1−SE)f(s) + ηE(t)
dSI/dt = −αSI + (1−SI)f(s) + ηI(t)
dSC/dt = −αSC + (1−SC)f(s) + ηC(t)
ηX(t) = (1/√NX)·√[αSX + f(s)(1−SX)]·ξX(t)
Each population has the same structure as the original Wilson-Cowan equation. The only difference is that the common input s now includes C's contribution wC,net·SC.
Section 3 Summary
- Three state variables: k (E active), l (I active), m (C active)
- Same transition structure: Deactivation at rate α, activation at rate f(s)
- Symmetric input: All populations receive the same input (like de Candia)
- Extended input: s now includes +wC,net·SC
- Three noise terms: Each population has its own 1/√NX noise