Simultaneous Firing & Connection Formation

• Simultaneous firing and connection formation are mechanisms where excitable nodes synchronously activate while dynamically rewiring their interconnections.
• Mathematical models employing FitzHugh–Nagumo dynamics and time-varying adjacency matrices elucidate the impact of rewiring rates on network synchrony.
• Experimental protocols, including single-neuron mimicry and artificial axon setups, validate synchronization metrics and plasticity rules in controlled environments.

Simultaneous firing and connection formation refer to the processes by which excitable elements—biological or artificial—generate synchronized activity, and how these patterns are shaped or enabled by the evolving topology of inter-element connections. In neurobiology, these dynamics underlie phenomena such as synchrony, oscillations, and plasticity in neural networks. In synthetic systems and mathematical models, these mechanisms illuminate the interplay between network connectivity, node excitability, and emergent collective behaviors.

1. Dynamical Models of Simultaneous Firing

Simultaneous firing in sparsely connected ensembles can be rigorously studied using systems of coupled excitable nodes, such as FitzHugh–Nagumo elements. Each node is modeled by a fast activator $x_j(t)$ and a slow inhibitor $y_j(t)$, with dynamics: $$\epsilon\,\dot x_j = x_j - \frac{x_j^3}{3} + y_j + I, \qquad \dot y_j = a - x_j,$$ where $\epsilon \ll 1$ sets the time-scale separation, $a > 1$ ensures excitability, and $I$ is a constant bias. Nodes are coupled via a time-dependent adjacency matrix $A_{ij}(t)$ encoding the instantaneous network: $$\epsilon\,\dot x_j = x_j - \tfrac{x_j^3}{3} + y_j + k\sum_{i=1}^N A_{ji}(t)\bigl(x_i - x_j\bigr), \quad \dot y_j = a - x_j.$$ Instantaneous degree is strictly constrained: $\sum_{i=1}^N A_{ij}(t)\leq 1$ for all $j$, so that each node is connected to at most one partner at a given time. The coupling strength $k$ is typically set to unity for simulation (Tessone et al., 2012).

2. Network Rewiring and Connection Formation Protocols

Two primary schemes realize time-evolving networks with degree $\leq 1$:

• Fixed Matching: At all times, exactly $N/2$ undirected links ensure that each node has one partner. At total rate $\lambda N$, pairs of links $(j,j^*), (k,k^*)$ are randomly selected, and endpoints are swapped to form $(j,k), (j^*,k^*)$.
• Underlying Substrate: A substrate graph $G$ with $L$ possible edges is defined. Nodes can activate dormant edges at rate $\lambda$, deactivating any pre-existing edge to maintain degree $\leq 1$.

The instantaneous adjacency $A_{ij}(t)$ is symmetric with row sums $\leq 1$. The reconnection rate $\lambda$ governs the temporal statistics of partner switching, hence the pattern formation dynamics (Tessone et al., 2012).

In experimental neuroscience, "single-neuron mimicry" allows emulation of a full $N$-node recurrent network using one living neuron. For each network "snapshot," a node's recent inputs are precisely replayed on a biological cell, and its firing or failure is mapped to the network's evolution. Connectivity is encoded in the matrix $W_{ij}$ (weights) and $\tau_{ij}$ (delays), dictating which virtual nodes stimulate one another, and at which times (Goldental et al., 2016).

3. Synchronization Metrics and Order Parameters

Synchrony is quantified via several order parameters:

• Kuramoto Order Parameter ($\rho$): Using the geometric phase $\phi_j(t)=\arctan\bigl(y_j(t)/x_j(t)\bigr)$,

$$\rho(t)\,e^{i\Psi(t)} = \frac{1}{N}\sum_{j=1}^N e^{i\phi_j(t)};\quad \rho = \langle\rho(t)\rangle_t.$$

Static phase coherence corresponds to $\rho\approx 1$.

• Shinomoto–Kuramoto Parameter ($\zeta$): Measures dynamic, collective pulsations,

$$\zeta = \Bigl\langle \bigl|\rho(t)e^{i\Psi(t)} - \langle\rho(t)e^{i\Psi(t)}\rangle_t\bigr| \Bigr\rangle_t,$$

which is nonzero only if the center of mass of phases rotates around the circle.

• Mean Firing Current ($J$):

$$J = \Bigl\langle \Bigl|\frac{1}{N}\sum_{j=1}^N \dot x_j(t)\Bigr|\Bigr\rangle_t,$$

signaling activity regardless of synchrony.

These parameters reveal regimes of quiescence ($\rho\approx1$, $\zeta\approx0$, $J\approx0$), partial synchrony, and full, dynamic global firing (Tessone et al., 2012).

4. Mechanisms and Windows for Globally Synchronized Firing

The occurrence of fully synchronized firing (all elements spike in unison) is contingent on the interplay between network evolution rate $\lambda$ and coupling $k$:

• Slow Rewiring ($\lambda\ll\lambda_{c1}$): Links persist so long that nodes always relax to rest before partners change; no excitation propagates.
• Intermediate Rewiring ($\lambda_{c1}<\lambda<\lambda_{c2}$): When typically $$\lambda \sim (\tau_{\rm left}+\tau_{\rm right})^{-1}$$, nodes near rest can suddenly couple to a firing partner, receiving an "impulsive kick" sufficient to trigger their own excursion. This transient drives a spatial-temporal cascade, rapidly synchronizing the entire network. Avalanche-like propagation is observed, with a bell-shaped $\zeta(\lambda)$ and $J(\lambda)$ profile peaking in this window.
• Fast Rewiring ($\lambda\gg\lambda_{c2}$): Fluctuating couplings self-average; nodes experience only the static, ensemble-mean field and settle to a quiescent fixed point.

No synchrony emerges for $k<k_c \approx 0.8$ at any $\lambda$ (Tessone et al., 2012).

5. Connection Formation, Plasticity, and Experimental Emulation

In engineered or model networks, connection formation is studied both via explicit protocol and via emulation:

• Directed Graphs and Delays: The topology is set by $W_{ij}$ and delays $\tau_{ij}$. Firing events propagate according to

$$I_i(t)=\sum_{j=1}^N\sum_{k} W_{ij}\,\delta(t-t_j^{(k)}-\tau_{ij})+\eta_i(t),$$

with $t_j^{(k)}$ the $k$th spike of node $j$.

• Plasticity Update Rules: Connection strengths can be dynamically updated after firing events using prespecified functions (e.g., $$\Delta W_{ij}=\mathcal{F}(W_{ij},\{t_j^{(k)},t_i\})$$, enabling simulation of mechanisms such as spike-timing-dependent plasticity (STDP)).
• Experimental Single-Neuron Mimicry: The single-neuron protocol supports the full software updating of $W_{ij}$ and $\tau_{ij}$ post-hoc, allowing studies of both fixed and evolving connection topologies in real time (Goldental et al., 2016).

6. Artificial Excitable Systems and Coincidence Firing

Artificial axon systems provide hardware for precise control and study of simultaneous firing and connection formation:

• Node Construction: A lipid bilayer supports a population of KvAP ion channels between reservoirs with K$^+$ gradients, establishing a Nernst potential. Electronic current-limited voltage clamps (CLVC) maintain resting potential until input stimuli overcome thresholds.
• Coincidence Detector Protocol: Subthreshold pulses delivered singly do not trigger firing; two temporally coincident pulses within 50–100 ms sum their conductances, reliably producing an action potential. This realizes an AND-type logic operation.
• Synaptic Coupling: Pre-synaptic voltage $V_1(t)$ drives a current $I_s(t)=\alpha V_1(t)\Theta(V_1(t))$ into the post-synaptic node. Action potentials in axon 1 induce firing in axon 2, capturing pre/post synaptic relationships and enabling programmable network motifs (Vasquez et al., 2017).

7. Emergent Oscillatory Dynamics and Experimental Outcomes

Both modeling and experimental emulation reveal multiple collective firing patterns:

• Gamma ($\gamma$) Oscillations: In uniform-delay recurrent networks, loop periodicity locks firing events into volleys at frequency $f_\gamma\approx 1/\tau$ ($\tau$ the synaptic delay), yielding rhythms in the 50–100 Hz range.
• Delta ($\delta$) Oscillations: Global firing rate envelopes fluctuate at 1–5 Hz, emerging from recurrency, saturation effects, and stochastic noise. These phenomena are observed both in software-driven networks and in experimental single-neuron mimicry, and are measurable as population raster plots with fast carriers modulated by slow envelopes (Goldental et al., 2016).
• Artificial Systems: Firing rate in artificial axons increases linearly with injected current up to a threshold, after which action potentials are reproducibly generated given sufficient current or coincident pulses. Synaptic efficacy is quantified by delay and magnitude of induced firing in coupled axons (Vasquez et al., 2017).

System Type	Firing Synchronization Mechanism	Connection Adaptation
FitzHugh–Nagumo	Intermediate-rate random partner rewiring	Implicit; formation via rewiring
Single-neuron mimicry	Sequence replay, inputs emulate full network	Software-driven updating of weights
Artificial axon	Coincidence and synaptic current injection	Hardwired electronic circuits

These advances establish that collective, synchronous firing can be generated and studied even in disconnected or minimally connected networks, with network evolution and input timing providing sufficient drive for emergent organization. Furthermore, experimental and theoretical techniques now enable detailed, controlled studies of both the emergence of synchrony and the effect of network wiring—fixed or plastic—on those patterns (Tessone et al., 2012, Goldental et al., 2016, Vasquez et al., 2017).