Data_Sheet_1_Noise-induced synchrony of two-neuron motifs with asymmetric noise and uneven coupling.pdf

Synchronous dynamics play a pivotal role in various cognitive processes. Previous studies extensively investigate noise-induced synchrony in coupled neural oscillators, with a focus on scenarios featuring uniform noise and equal coupling strengths between neurons. However, real-world or experimental settings frequently exhibit heterogeneity, including deviations from uniformity in coupling and noise patterns. This study investigates noise-induced synchrony in a pair of coupled excitable neurons operating in a heterogeneous environment, where both noise intensity and coupling strength can vary independently. Each neuron is an excitable oscillator, represented by the normal form of Hopf bifurcation (HB). In the absence of stimulus, these neurons remain quiescent but can be triggered by perturbations, such as noise. Typically, noise and coupling exert opposing influences on neural dynamics, with noise diminishing coherence and coupling promoting synchrony. Our results illustrate the ability of asymmetric noise to induce synchronization in such coupled neural oscillators, with synchronization becoming increasingly pronounced as the system approaches the excitation threshold (i.e., HB). Additionally, we find that uneven coupling strengths and noise asymmetries are factors that can promote in-phase synchrony. Notably, we identify an optimal synchronization state when the absolute difference in coupling strengths is maximized, regardless of the specific coupling strengths chosen. Furthermore, we establish a robust relationship between coupling asymmetry and the noise intensity required to maximize synchronization. Specifically, when one oscillator (receiver neuron) receives a strong input from the other oscillator (source neuron) and the source neuron receives significantly weaker or no input from the receiver neuron, synchrony is maximized when the noise applied to the receiver neuron is much weaker than that applied to the source neuron. These findings reveal the significant connection between uneven coupling and asymmetric noise in coupled neuronal oscillators, shedding light on the enhanced propensity for in-phase synchronization in two-neuron motifs with one-way connections compared to those with two-way connections. This research contributes to a deeper understanding of the functional roles of network motifs that may serve within neuronal dynamics.</p

Noise induces or modifies bursting.

<p>(<b>A</b>) Without noise, bursts are as shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g006" target="_blank">Figure 6B</a> but here, under the influence of noise (σ = 0.1 µA/cm²), the bursting periods are random. Right (here and in B): an expanded section showing STOs induced by noise via coherence resonance. (<b>B</b>) Without noise the system is in the bistability regime as shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g006" target="_blank">Figure 6C</a>, but here, noise (σ = 0.4 µA/cm²) causes <i>V_m</i> to randomly visit quiescent or repetitive firing states, hence producing noisy bursts.</p

Partial Nav-CLS in a population and apparently first order activation kinetics.

<p>Equilibrium activation values (<i>m³</i>, and <i>m</i> ) of co-existing intact Nav and CLS Nav channels plotted as a function of <i>V_m</i>. Black solid and black dashed line represent <i>m³</i> for intact and 20 mV CLS channels with <i>f</i> = 1. For a membrane with large Nav-CLS injury given by: <i>LS_i</i> = [0,20]mV and <i>f_i</i> = [0.3,0.7], the gray bold line denotes . Red line: <i>m</i> of the intact channels.</p

Classes of Nav-CLS-induced activities with two injured and one intact <i>g_Na</i> population.

<p>(<b>A</b>) Membrane excitability map of injured node with <i>LS_i</i> = [26.5,2,0]mV. White, pink, green and blue regions represent different stable state activities: quiescence, bursts, tonic firing, and bistability between quiescence and tonic firing. Here the fraction of intact <i>g_Na</i> is <i>f₃</i> = 1−<i>f₁−f₂</i>. (<b>B</b>) and (<b>C</b>), typical <i>V_m</i> trajectories when the system is in bursting and bistability regimes, respectively.</p

Steady-states for intact nodes (<i>V_m</i> = −59.9 mV) as <i>I_maxpump</i> varies (units as in Table 1).

<p>Steady-states for intact nodes (<i>V_m</i> = −59.9 mV) as <i>I_maxpump</i> varies (units as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-t001" target="_blank">Table 1</a>).</p

Burst dynamics explained with three-dimensional <i>V_m</i> trajectories and bifurcation diagrams

<p>(for bursts in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g004" target="_blank">Figure 4A</a> with <i>I_maxpump</i> = 95 µA/cm²). Note: x-axis scales are different is each plot. (<b>A</b>) The first burst of spikes in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g004" target="_blank">Figure 4A</a> plotted as a function of time are now plotted in 3-D as a function of <i>E_Na</i> and <i>E_K</i> (blue arrow: direction of <i>V_m</i> trajectory). (<b>B</b>) Bifurcation diagram for fixed <i>E_K</i> and <i>I_pump</i> as per <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g004" target="_blank">Figure 4A</a> pink star; <i>E_Na</i> is the slowly varying parameter. Lines and points are labeled (see also abbreviation list) and have the same meanings in (C). For <i>E_Na</i> at its <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g004" target="_blank">Figure 4A</a> pink star value (40.2 mV), the only stable solution is a periodic orbit (pink oval), with each cycle corresponding to a spike (in a burst). The oval-loop symbolizes one cycle of <i>V_m</i> oscillation at fixed <i>E_Na</i>. During a burst the <i>E_Na</i> decline shifts orbits leftward into the bistability regime (pink area) where there exist two stable solutions: a limit cycle and a stable fixed point. (<b>C</b>) Bifurcation diagram for <i>E_K</i> and <i>I_pump</i> fixed as per <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g004" target="_blank">Figure 4A</a> green dot. For <i>E_Na</i> at its <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g004" target="_blank">Figure 4A</a> green dot value (37.27 mV), the system (large green dot) is within the bistability regime (green area). <i>V_m</i>, attracted by this fixed point, has STOs (drawn as the green loop) until <i>E_Na</i> increases and superthreshold-oscillations (spikes) return. The PD region corresponds to period-doubling bifurcations. (<b>D</b>) Two-parameter phase diagram for <i>E_K</i> and <i>E_Na</i>. Pink solid and dashed curves represent LP (saddle-node bifurcation) and HB, respectively, when <i>I_pump</i> is fixed as in B. The green solid, dashed, and dash-dotted curves represent LP, HB and PD, respectively, when <i>I_pump</i> is fixed as in C. With varying <i>I_pump</i> the bistability regime shifts from the pink to the green area (the zone with both colors is the overlap of these two areas). The gray area between two green dash-dotted curves is a zone with PD bifurcations. The black loop shows <i>E_K</i> and <i>E_Na</i> orbits during a burst.</p

Ectopic steady-state <i>V_m</i> excursions are tuned by <i>I_pump</i>.

<p>(<b>A</b>) From the spontaneous tonic firing regime of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g009" target="_blank">Figure 9</a> in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi.1002664-Boucher1" target="_blank">[11]</a>, a random point (<i>LS_i</i> = [0,15]mV, <i>f_i</i> = [0.5,0.5]) is selected; as illustrated, <i>V_m</i> oscillation amplitude varies as <i>I_maxpump</i> is set at 30 and 95 µA/cm². (<b>B</b>) <i>I_pump</i> changes from a constant 10 µA/cm² (associated with a small periodic <i>V_m</i> fluctuation) to a fluctuating 23±0.5 µA/cm² (associated with a train of APs). (<b>C</b>) Corresponding total <i>I_Na</i>. (<b>D</b>) With growing <i>I_maxpump</i>, voltage excursions (blue) increase and their oscillation frequencies (red) decrease.</p

Nav-CLS induced spontaneous activities of injured node: steady state with fixed <i>E_Ion</i>, tonic spiking, tonic subthreshold oscillations (STOs).

<p>By setting <i>I_maxpump</i>, <i>g_Naleak</i>, and <i>g_Kleak</i> to zero and eliminating <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi.1002664.e029" target="_blank">Eqs (11)</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi.1002664.e034" target="_blank">(12)</a>, the <i>E_Ion</i> are artificially maintained at fixed values. Three sets of values are considered: left column, <i>E_Na</i> = 50 mV, <i>E_K</i> = −77 mV; middle column: <i>E_Na</i> = 42 mV, <i>E_K</i> = −77 mV; right column: <i>E_Na</i> = 42 mV, <i>E_K</i> = −71 mV. For the first 3 rows, Nav-CLS have Gaussian distributions (mean±SD): (<b>A,B,C</b>) 1.3±0.4 mV; (<b>D,E,F</b>) 8±1 mV; (<b>G,H,I</b>) 15±1 mV. For the last rows,(<b>J,K,L</b>) bifurcation diagrams (solution of <i>V_m</i> in terms of <i>LS</i>) are plotted, and there, for computational tractability, single <i>g_Na</i> populations (i.e. <i>f</i> = 1, no Gaussian “smears”) are used. As labeled, the solid line, dashed line, filled dots, open dots and “HB” respectively denote: stable fixed point (i.e. resting potential (RP)), unstable fixed point, stable limit cycle (SLC, i.e., tonically firing APs), unstable limit cycle (ULC) and Hopf bifurcation point (HB). When <i>E_Na</i> alone changes (<b>J</b> to <b>K</b>), the bifurcation structure shows only slight changes in the amplitude of SLC and the locations of subcritical HB on both ends. When <i>E_K</i> changes (<b>K</b> to <b>L</b>), the HBs on both ends shift to the left (i.e. towards relatively smaller <i>E_Na</i>) and the previously subcritical HB (right side) becomes supercritical. Across the 3 columns, note that if the system did have pumps, interactions between the <i>E_Ion</i> and <i>I_pump</i> would continually and slowly change <i>E_Ion</i> thereby repeatedly evoking these activity patterns.</p

Transitions between STO and burst behaviors.

<p>(<b>A</b>) Upper: <i>V_m</i> (black solid line) at an injured node and the varying <i>E_Ion</i> (<i>E_Na</i> and <i>E_K</i>; blue dotted lines). Three <i>g_Na</i> populations are used: <i>LS_i</i> = [0,2,26.5]mV and <i>f_i</i> = [0.72,0.08,0.2] with <i>Vol_i</i> = <i>Vol_o</i> = 10⁻¹⁵ m³. Initiation and termination times of a burst of spikes (pink star, green dot, respectively) are used in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi-1002664-g005" target="_blank">Figure 5</a>. Lower: corresponding Na⁺ currents, as labelled. (<b>B</b>) Equilibrium values of Nav channel activation and inactivation variables, <i>m_i</i> and <i>h_i</i>. (<b>C</b>) The steady-state open probabilities, : with a mild injury <i>g_window</i>(V) magnitude is slightly less at 0 mV (vertical line and circles) but much enlarged at voltages near the normal RP (−65.5 mV for fixed [ion] condition). (<b>D</b>) Another example: <i>LS_i</i> = [0,2,20]mV and <i>f_i</i> = [0.72,0.08,0.2] and <i>Vol_i</i> = <i>Vol_o</i> = 3×10⁻¹⁵ m³, with expanded detail showing STOs. Note the difference of time scales in (A) and (D), reflecting the fact that a 3-fold lower axonal surface-to-volume ratio in (D) slows the rate of [ion] changes.</p

Schematic of a mechanically-injured node of Ranvier

<p>depicted with a mix of intact-looking and severely-blebbed axolemma (as labelled) such as seen in transmission electromicrographs of stretch-injured optic nerve nodes <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi.1002664-Maxwell1" target="_blank">[2]</a>. In pipette aspiration bleb injury, the cortical actomyosin-spectrin skeleton progressively detaches <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002664#pcbi.1002664-Sheetz1" target="_blank">[6]</a>. Our model considers a node as one equipotential compartment in which actual spatial arrangements of pumps and channels are irrelevant. However, the fraction of Nav channels in the injured portion of the membrane, along with the severity of their gating abnormality, are model parameters.</p