Excitation–Inhibition Network with Homeostatic Plasticity

• Excitation–inhibition networks are neural systems defined by balanced excitatory and inhibitory interactions, shaped by homeostatic plasticity to maintain stable activity.
• They employ synaptic scaling and Hebbian updates to modulate synaptic strengths, ensuring robustness against runaway excitation and sustaining memory traces.
• The model uses dynamic up-/down-state transitions and a control parameter R to reproduce power-law avalanche statistics and critical neural dynamics.

An excitation–inhibition network with homeostatic plasticity is a neural system in which excitatory and inhibitory neurons dynamically interact on complex network topologies, while synaptic strengths are adaptively regulated by homeostatic feedback mechanisms that maintain functional stability and control network excitability. These networks are central to the regulation of critical phenomena such as neuronal avalanches, the emergence of up/down states, and the self-organization of temporal dynamics and memory in cortex-like architectures. Homeostatic mechanisms, implemented via synaptic scaling or local adaptation, ensure that excitation and inhibition remain balanced even in the presence of plastic changes to the network, thus stabilizing collective dynamics and permitting the flexible temporal organization observed in cortical activity.

1. Neuronal Model and Network Architecture

Individual neurons are represented by membrane potentials $v_i$ evolving on a scale-free network. Each neuron fires when $v_i$ reaches a threshold $v_{max}$. Upon firing, neuron $i$ emits a charge $q_i \propto v_i k_{out,i}$, which is distributed to postsynaptic targets according to synaptic weights $g_{ij}$, with the update: $$v_j(t+1) = v_j(t) \pm \frac{q_i(t)}{k_{in,j}} \cdot \frac{g_{ij}(t)}{ \sum_{k} g_{ik}(t) }$$ The sign is positive for excitatory, negative for inhibitory synapses. This mechanism embeds the network topology and current synaptic states directly into the synaptic transmission process, explicitly coupling excitation and inhibition via the network architecture.

2. Homeostatic Synaptic Plasticity

After each avalanche of activity, synaptic strengths are governed by a combination of Hebbian modifications and global homeostatic scaling:

• Hebbian potentiation: For active synapses in an avalanche,

$$g_{ij}(t+1) = g_{ij}(t) + \frac{ v_j(t+1) - v_j(t) }{ v_{max} }$$

• Homeostatic synaptic scaling: Inactive synapses are depressed by the average increment $\Delta g$:

$$\Delta g = \frac{ \sum_{(ij, t)} \delta g_{ij}(t) }{N_b }$$

where $N_b$ is the total synapse count.

This process increases the strength of synapses activated during avalanches, while downscaling inactive synapses, preventing runaway excitation or silencing, and maintaining the balance of synaptic efficacy across the network. The combination of these rules forms a homeostatic regulatory mechanism essential for sustaining criticality.

3. State-Dependent Modulation: Up- and Down-States

The temporal structure of network activity is segmented by alternations between up-states (high excitability) and down-states (low excitability), dynamically regulated according to avalanche size:

• Up-state: For small avalanches ($s_{\Delta v} < s_{\Delta v}^{min}$), recently active neurons are reset to a depolarized state:

$$v_i = v_{max} \left( 1 - \frac{ s_{\Delta v} }{ s_{\Delta v}^{min} } \right)$$

Neurons remain close to threshold, promoting high excitability and correlated avalanche sequences.

• Down-state: For large avalanches ($s_{\Delta v} > s_{\Delta v}^{min}$), active neurons are hyperpolarized:

$v_i = v_i - h \delta v_i$

where $h > 0$ determines the inhibitory strength. This results in an enforced refractory period and temporarily reduces excitability.

The alternation between up and down states captures essential features of cortical dynamics and is pivotal for the emergence of critical-like statistics and memory-dependent waiting-time distributions.

4. Control Parameter and Excitation–Inhibition Balance

The global balance between excitation and inhibition is controlled by a non-dimensional parameter: $R = \frac{h}{ s_{\Delta v}^{min} }$ $R$ encapsulates the relationship between hyperpolarization strength ($h$) and the sensitivity threshold separating up- and down-states ($s_{\Delta v}^{min}$). Network simulations demonstrate that precise tuning of $R$ is required to maintain the empirically observed non-monotonic, power-law waiting time distributions characteristic of neuronal avalanches. Decreasing $R$ extends up-states (excess excitation), while increasing $R$ shortens up-states, shifting dynamics toward enhanced inhibition and longer quiescent intervals.

5. Temporal Organization, Correlations, and Memory

Homeostatic plasticity and state-dependent potential resets together generate a rich temporal organization of neural avalanches:

• Up-states: Produce waiting-time distributions with power law regimes, indicating clustering of avalanches and preservation of network “memory” (neurons remain near threshold).
• Down-states: Lead to bell-shaped waiting-time distributions, reflecting desynchronization and erasure of network memory owing to hyperpolarization.
• Non-monotonic distribution: Resulting from up/down alternation, the waiting-time distribution is non-monotonic, recapitulating experimental findings.

Thus, the network’s memory of past activity and its capacity for dynamic excitability is jointly encoded by plasticity-modulated synaptic strengths and the interplay of excitation–inhibition balancing mechanisms.

6. Mathematical Summary of Key Update Rules

Mechanism	Update Formula	Role
Membrane potential update	$v_j(t+1) = v_j(t) \pm \ldots$	Encodes E/I interaction, network topology
Hebbian potentiation	$$g_{ij}(t+1) = g_{ij}(t) + [v_j(t+1) - v_j(t)]/v_{max}$$	Synaptic strengthening along active pathways
Homeostatic scaling	$\Delta g = \frac{ \sum \delta g_{ij}(t) }{ N_b }$	Global normalization to maintain overall excitability
Up-state reset	$$v_i = v_{max} (1 - s_{\Delta v} / s_{\Delta v}^{min})$$	Promotes high excitability, clustering
Down-state hyperpolarization	$v_i = v_i - h \delta v_i$	Enforces refractory period, erases memory
Excitation–inhibition control $R$	$R = h / s_{\Delta v}^{min}$	Sets balance, tunes criticality

7. Broader Implications and Context

This model demonstrates how a biologically grounded interplay of network-level homeostatic plasticity, local Hebbian modifications, and dynamic switches between up-/down-states can account for the empirically observed temporal organization and avalanche statistics in cortical networks (Lombardi et al., 2012). By reducing balance control to a single dimensionless parameter $R$, the model provides a mechanistic bridge connecting the maintenance of excitability, the embedding of memory, and the emergence of criticality—without the need for precise global fine-tuning. Such networks are robust to fluctuations, adaptive to past activity, and capable of self-organizing their temporal structure to optimize information transmission under physiological constraints. These findings offer a theoretical framework for linking cellular plasticity mechanisms to emergent, system-level properties of brain dynamics.