Neural Plasticity Dynamics

• Neural plasticity dynamics are the study of changes in synaptic connectivity driven by precise spike timing, delay heterogeneity, and STDP.
• Mathematical models integrate Hodgkin–Huxley dynamics with time-delayed synaptic transmission to reveal distinct synchronization regimes.
• Dynamic feedback loops between activity and connectivity sculpt modular brain networks, aligning functional synchrony with emerging structural motifs.

Neural plasticity dynamics encompass the evolution of synaptic connectivity in response to the timing and patterns of neuronal activity, with effects that operate over multiple timescales and are tightly coupled to circuit topology and network-level phenomena. In theoretical and computational neuroscience, the paper of these dynamics aims to bridge the microstructure of synaptic updates and the mesoscopic or macroscopic organizational principles of functional circuits.

1. Mathematical Formulation of Neural Plasticity Dynamics

Plasticity dynamics are governed by interaction between intrinsic neuronal dynamics, synaptic transmission (with axonal/dendritic delays), and plastic adaptation rules typically based on the precise timing of pre- and postsynaptic spikes. In conductance-based models, such as the Hodgkin–Huxley network considered in (Protachevicz et al., 2022), each neuron's membrane voltage $V_i$ and associated gating variables evolve according to:

$$\begin{aligned} C\,\frac{dV_i}{dt} &= I_{i} - g_K n^4_i (V_i - E_K) - g_{Na} m^3_i h_i (V_i - E_{Na}) - g_L (V_i - E_L) \ &\quad + (V_r^+ - V_i) \sum_{j=1}^N g_{ij} f_j(t- \tau_{ij}), \end{aligned}$$

$\tau_s \frac{df_i}{dt} = -f_i,$

where $f_i$ signals postsynaptic conductance in response to spiking.

The network is modular: $N=400$ neurons are divided into $S=4$ subnetworks, each fully internally connected, with sparse external links ($p_{int}=1$, $p_{ext}=0.05$). Delays differentiate intra- and inter-subnetwork signaling: $\tau_{ij} = \tau_{int}$ if $i$ and $j$ share a subnetwork, $\tau_{ij} = \tau_{ext}$ otherwise, with $\tau_{int}$ and $\tau_{ext}$ swept in simulation across physiologically plausible ranges.

Plasticity is implemented via classical spike-timing–dependent plasticity (STDP) with temporally asymmetric learning windows:

$$\Delta g_{ij} = \begin{cases} A_1 e^{-\Delta t_{ij}/\tau_1} & \Delta t_{ij} \geq 0 \ - A_2 e^{\Delta t_{ij}/\tau_2} & \Delta t_{ij} < 0 \end{cases}, \quad g_{ij} \leftarrow g_{ij} + G \Delta g_{ij}$$

where $$\Delta t_{ij}=t_i^{\mathrm{post}} - t_j^{\mathrm{pre}}$$; $A_1=1$, $A_2=0.5$, $\tau_1=1.8\,\text{ms}$, $\tau_2=6\,\text{ms}$, and $G=10^{-5}$ mS/cm$^2$. Synaptic weights are strictly bounded.

2. Emergence of Delay-Induced Synchronization Regimes

The combination of delay heterogeneity and STDP results in rich collective dynamics, with distinct synchronous regimes depending on the intra- and inter-subnetwork delays:

• For small $\tau_{int}\approx 0$ ms and increasing $\tau_{ext}$:
  • $\tau_{ext}=0$ ms yields global synchrony (1-group, $R^1 \approx 1$).
  • $\tau_{ext}\approx 4$ ms induces two-group anti-phase synchrony (dominant $R^2$).
  • $\tau_{ext}\approx 6$ ms produces four-group symmetry with $\pi/2$ phase separation (dominant $R^4$).
  • $\tau_{ext} \gg$ mean ISI eventually returns to global synchrony.
• Elevation of $\tau_{int}$ ($3$–$6$ ms) breaks within-subnetwork coherence, diminishing all synchrony indices.

Phase relationships are quantified using global and subnetwork Kuramoto order parameters, as well as $m$-th order generalized order parameters $R^m$, measuring the degree of $m$-fold phase-symmetric clustering in the spike timing distribution.

3. Mechanisms Linking Functional and Structural Plasticity

Plasticity dynamically sculpts network topology to reinforce delay-favored synchronous patterns:

• In the one-group case, STDP yields a hierarchical chain, potentiating all inter-subnetwork connections consistent with the leading phase order.
• In the two-group regime, intra-group connections are potentiated, while anti-phase links are systematically depressed, forming two disconnected blocks.
• The four-group regime produces potentiation in a cyclic ("ring") topology, matching the phase-locked rotations driven by delays.
• At maximal delay ($\tau_{ext} \sim$ mean ISI), STDP homogenizes weights, yielding uniform potentiation.

Empirically, the regime boundaries in delay space are modified by plasticity, with the domains supporting symmetric multi-clustered states (e.g., $m=2,4$) expanding due to delay-activity-driven reinforcement.

4. Analytical and Simulation Tools for Characterizing Dynamics

The paper employs several quantitative tools:

Measure	Mathematical Definition/Usage	Role in Analysis
Kuramoto parameter ($R_T$)	$$R_T(t) = \left| \frac{1}{N}\sum_j e^{i\phi_j(t)} \right|$$	Global synchrony index
$m$-th moment order ($R^m$)	$$R^m(t) = \left| \frac{1}{N}\sum_j e^{im\phi_j(t)}\right|$$	Detects multi-cluster patterns
Structural connectivity	Averaged $g_{ij}$ between/within subnetworks	Reveals plasticity-driven motifs
Graphical summaries	"Strongest edges" diagrams post-plasticity	Schematic of dominant motifs

Structural-functional alignment is established by comparing weight matrices at fixed time (post-plasticity) against the phase-synchrony regime maps throughout and after adaptation.

5. Dynamical Self-Organization and Feedback Loops

Delays define preferred phase-lags between subnetworks, with STDP reinforcing causally-coherent links ($\Delta t$ positive, in-phase) and weakening anti-causal or anti-phase connections (negative or large $\Delta t$). This reciprocal feedback—delays bias firing patterns, which in turn refashion connectivity via plasticity—produces dynamic self-organization, locking the system into robust phase-locked states that are mirrored in both emergent function and structure.

Specifically, function$\to$structure mapping occurs as spike-timing statistics sculpt connectivity. Structure$\to$function mapping implies that the adapted topology directly promotes the synchronization motif that generated it. A coupled loop ensues, with functional patterns and structural motifs converging to a stable, mutually reinforcing configuration.

6. Biological Implications and Generalization

The results generalize to modular brain architectures: short-range delays within regions promote local coherence, while long-range inter-areal delays support flexible, delay-tuned phase relationships or segregation between functional clusters. The mechanism described permits the linking of repeated phase-locked activity to the consolidation of structural pathways, with implications for memory formation, dynamic routing, and the emergence of modular brain organization.

Proposed model extensions include:

• Incorporation of inhibitory neurons and inhibitory plasticity (e.g., inhibitory STDP).
• Axonal velocity heterogeneity.
• Burst-dependent plasticity rules.
• Homeostatic regulation and activity-dependent synaptic scaling.
• Structural rewiring mechanisms beyond weight modulation.

7. Equivalence of Network Function and Structure

The principal finding is the dynamical equivalence of function (synchrony pattern) and structure (connectivity motif): the same delays and plasticity rules that drive the emergence of spatiotemporal firing patterns simultaneously imprint corresponding structural motifs in the recurrent connectivity. The adaptation of network topology to sustained activity patterns ("topology emerges from behaviour") and the reciprocal tuning of network dynamics by the revised architecture ("behaviour emerges from topology") characterize a complex feedback process in which function and structure are not separable but co-evolving entities. This property provides a rigorous theoretical underpinning for the functional modularity and phase-coherence dynamics observed in real neuronal circuits (Protachevicz et al., 2022).