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Phase model analysis of M-current effects on neural synchrony in hippocampal networks. Theoretical framework linking acetylcholine neuromodulation to neural assembly formation via phase reduction and cluster synchronization.
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| name | phase-model-m-current-hippocampal-synchrony |
| description | Phase model analysis of M-current effects on neural synchrony in hippocampal networks. Theoretical framework linking acetylcholine neuromodulation to neural assembly formation via phase reduction and cluster synchronization. |
| keywords | ["phase model","M-current","neural synchrony","hippocampal networks","neural assemblies","acetylcholine","memory consolidation","cluster solutions","dynamical systems"] |
| version | 1.0.0 |
| arxiv_id | 2606.12684 |
| authors | Megha Manoj, Sue Ann Campbell |
| published | 2026-06-10T00:00:00.000Z |
| categories | ["q-bio.NC","math.DS"] |
This paper presents a one-dimensional phase model reduction to analyze how M-current (slow, voltage-dependent, non-inactivating potassium current) affects neural synchrony in hippocampal networks, providing a mechanistic explanation for acetylcholine's bidirectional role in memory encoding vs consolidation.
Key Innovation: Phase model reduction + cluster solution analysis → predicts synchronization states under different ACh levels
Core Question: How does acetylcholine modulation of M-current regulate neural assembly formation through synchrony?
Step 1: Reduce 2-neuron coupled system to 1D phase model
# Phase reduction: θ = θ₀ + ω·t + H(θ_other - θ)
# H: interaction function (phase coupling)
# M-current: I_M = g_M·m·(V - E_K) # Slow K+ current
Key Parameter: M-current conductance g_M (downregulated by ACh)
Step 2: Analyze symmetric cluster solutions
g_M → full synchronization (single cluster)g_M → multiple stable cluster solutions (distinct assemblies)All-to-all globally homogeneous coupling
Symmetric distance-dependent coupling
Nearest-neighbours coupling
dθ/dt = ω + H(Δθ) = 0High ACh (memory encoding):
↓ M-current (g_M ↓)
↓ Adaptation
↓ Synchrony
↑ Multiple clusters (distinct assemblies)
↑ Memory encoding capacity
Low ACh (memory consolidation):
↑ M-current (g_M ↑)
↑ Adaptation
↑ Synchrony
↓ Full synchronization
↑ Consolidation (assembly merging)
Mathematical Result: For N-neuron network with all-to-all coupling:
Δθ_k = 2π·k/N_clustersH'(Δθ_k) < 0Key Insight: Number of stable clusters inversely related to g_M
| State | ACh Level | M-Current | Synchrony | Neural Assemblies | Memory Stage |
|---|---|---|---|---|---|
| Active exploration | High | Low | Low | Many (desynchronized) | Encoding |
| REM sleep | High | Low | Low | Many | Encoding |
| Quiet waking | Low | High | High | Few (synchronized) | Consolidation |
| SWS sleep | Low | High | High | Single cluster | Consolidation |
Full neuron model (with M-current):
C·dV/dt = -I_Na - I_K - I_M + I_syn + I_ext
dm/dt = (m_inf(V) - m) / τ_m(V) # M-current activation
Phase-reduced model:
dθ₁/dt = ω₁ + H(θ₂ - θ₁; g_M)
dθ₂/dt = ω₂ + H(θ₁ - θ₂; g_M)
Interaction function H(φ; g_M):
For N-cluster solution θ_k = 2πk/N:
# Stability matrix
J_ij = H'(Δθ_k) for i ≠ j
J_ii = -sum(H'(Δθ_k))
# Stable if all eigenvalues < 0
def compute_interaction_function(model_params, g_M):
"""
Compute phase interaction function H(φ) from neuron model.
Parameters:
- model_params: {g_Na, g_K, C, ...}
- g_M: M-current conductance
Returns:
- H(φ): phase coupling function
"""
# 1. Find limit cycle (periodic orbit)
V0, period = find_limit_cycle(model_params, g_M)
# 2. Compute phase response curve (PRC)
Z(φ) = compute_PRC(V0, model_params)
# 3. Compute synaptic interaction
I_syn(φ) = synaptic_current(φ)
# 4. Phase interaction: H(φ) = ∮ Z(φ)·I_syn(φ) dφ
H = integrate_PRC_synaptic(Z, I_syn)
return H
def find_cluster_solutions(N_neurons, H, g_M):
"""
Find stable symmetric cluster solutions.
Returns:
- cluster_sizes: [N_1, N_2, ...] stable cluster sizes
- stability: [True/False for each]
"""
# Try all possible cluster partitions
for N_clusters in range(1, N_neurons+1):
Δθ = 2π / N_clusters
# Check stability
eigenvalues = compute_stability_eigenvalues(H, Δθ, N_clusters)
if all(eig < 0 for eig in eigenvalues):
yield N_clusters, Δθ, True
Use Case: Predict optimal ACh level for encoding new memories
# For encoding: maximize number of stable clusters
optimal_g_M = minimize(
lambda g_M: -len(find_cluster_solutions(N, H, g_M)),
bounds=[0, g_M_max]
)
# Corresponds to high ACh level during active exploration
REM vs SWS: Simulate synchrony differences
# REM sleep (high ACh, low g_M)
clusters_REM = find_cluster_solutions(N, H, g_M=0.1) # Many clusters
# SWS sleep (low ACh, high g_M)
clusters_SWS = find_cluster_solutions(N, H, g_M=1.0) # Full sync
Therapeutic Application: Optimize ACh agonist/antagonist dosing
# Memory disorder treatment
# Increase ACh for encoding deficits
# Decrease ACh for consolidation deficits
def optimal_ach_dosing(memory_stage, deficit_type):
if memory_stage == 'encoding' and deficit_type == 'low':
return high_ach_target # Desynchronize
elif memory_stage == 'consolidation':
return low_ach_target # Synchronize
g_M via lightTrigger phrases:
This skill enables understanding how acetylcholine neuromodulation of M-current regulates neural synchrony and assembly formation in hippocampal networks, providing a theoretical foundation for memory encoding vs consolidation mechanisms.