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. 2024 Aug 9:4:1423023.
doi: 10.3389/fnetp.2024.1423023. eCollection 2024.

How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction

Lev A Smirnov  1 , Vyacheslav O Munyayev  1 , Maxim I Bolotov  1 , Grigory V Osipov  1 , Igor Belykh  2
Affiliations

How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction

Lev A Smirnov et al. Front Netw Physiol. .

Abstract

The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.

Keywords: Kuramoto model; integrate-and-fire models; network physiology; partial synchronization; synaptic activation; synchronization; theta neurons; time delay.

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Conflict of interest statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Figures

FIGURE 1
FIGURE 1
Synaptic dynamics St , defined by (Eq. 4), induced by presynaptic spikes pt , taking the form of (A,B) P∞t as defined by (Eq. 6) and (C,D) Pνt as defined by (Eq. 5) with ν=40 and linearly increasing phase θ1(t)=t (N=1) . The parameter τ=0.18 is used for all cases.
FIGURE 2
FIGURE 2
Characteristics of the synaptic dynamic profile for St (Eq. 4) as a function of q . (A) Peak latency, tm , (B) maximum value, Sm , and (C) full width at half maximum (FWHM) , induced by spikes Pνt with τ=0.1 and different ν (cyan markers – ν=10 , red markers – ν=100 , orange markers – ν=1000 ).
FIGURE 3
FIGURE 3
Regions of attractive (blue) and repulsive (red) coupling in the theta neuron model (Eqs 3, 4), corresponding to the KS model regions defined by (Eq. 29). The colors represent the coupling strength K sign as a function of the synaptic time constant τ and the common external input θ for a fixed q . (A) q=2 , (B) q=4 , and (C) q=12 . Other parameters: ϰ=−0.2π , ν=20 , η1=η2=⋯=ηN=η . The yellow points A , B , C , and D indicate the parameter values used for numerical simulations of Figures 6–9.
FIGURE 4
FIGURE 4
Coupling strength K (A) and Sakaguchi parameter α (B) as functions of synaptic adaptation/delay (parameter q ) and the finite pulse width (parameter ν ). The dependencies are calculated analytically via (Eq. 21) and (Eq. 27) for τ=0.15 , ϰ=−0.2π , η¯=2 . The coupling strength K increases as the pulse width decreases (via increasing ν ) and decreases as synaptic adaptation slows down and experiences larger time delays (via increasing q ). The Sakaguchi parameter α is highly sensitive to q and only weakly dependent on ν .
FIGURE 5
FIGURE 5
Dynamical equivalence between the theta neuron (Eqs 3, 4) and KS models (Eq. 20), demonstrated via the onset of full synchronization. (A) The evolution of the first |R1| (solid curves) and second |R2| (dashed curves) order parameters for the theta neuron (green curves) and KS model (red curves), including the transient period. Initial phases θn , n=1,…,N=21 are uniformly distributed over the interval [−π;π] . (B) The colors depict the phase differences θn−θ21 in the theta neuron model converging to imperfect full synchronization, subject to mismatched parameters ηn that are uniformly distributed on the segment η¯−δη/2;η¯+δη/2 , η¯=2.0 , δη=6×10−3 . (C) Comparison between the dynamics of the phase differences θn−θ21 for n=n1=17 (thick red curve) and n=n2=20 (thick blue curve) in the theta neuron model and θn−θ21 recalculated from phases φn using the relation (Eq. 7) for n=n1=17 (thin cyan curve) and n=n2=20 (thin red curve) in the KS model. Note the perfect alignment of the phase-difference dynamics in the two models. Parameters: q=2 , ϰ=−0.2π , τ=0.5 , ν=20 , η¯=2.0 correspond to point A on Figure 3.
FIGURE 6
FIGURE 6
Dynamical equivalence between the theta neuron (Eqs 3, 4) and KS models (Eq. 20), demonstrated via the onset of non-stationary generalized splay state with an oscillating |R1|≈0 . Notations are as in Figure 5. (A) The evolution of the first |R1| (solid curves) and second |R2| (dashed curves) order parameters for the theta neuron (green curves) and KS model (red curves), including the transient period. (B) The colors depict the phase differences θn–θ21 in the theta neuron model. (C) Comparison between the dynamics of the phase differences θn–θ21 for n = n1 = 17 (thick red curve) and n = n1 = 20 (thick blue curve) in the theta neuron model and θn–θ21 for n = n1 = 17 (thin cyan curve) and n = n1 = 20 (thin red curve) in the KS model. Mismatch parameters ηn are chosen from a uniform distribution η¯−δη/2;η¯+δη/2 , η¯=2.0 , δη=10−3 . Other parameters: N=21 , q=2 , ϰ=−0.2π , τ=0.8 , and ν=20 correspond to point B on Figure 3 and yield the frequency parameter Ω≈2.639 , calculated from (Eq. 22) [not shown].
FIGURE 7
FIGURE 7
Large-size networks. Dynamical equivalence between the theta neuron (Eqs 3, 4) and KS models (Eq. 20), demonstrated via the onset of full synchronization (A,B) and non-stationary generalized splay state with an oscillating |R1|≈0 (C,D) for N=100 (A,C) and N=1,000 (B,D). Notations are as in Figures 5, 6. Parameters: q=2 , ϰ=−0.2π , τ=0.5 , ν=20 , η¯=2.0 (A,B); q=2 , ϰ=−0.2π , τ=0.8 , and ν=20 (C,D).
FIGURE 8
FIGURE 8
Onset of full synchronization in the QIF (Eq. 1) and KS models (Eq. 20). (A) Firing rate and (B) firing times of QIF neurons (cyan curves and round markers) and oscillators of the KS model (black curves and cross markers) with the firing times recorded at θn(tf)=π . Each row in (B) represents the firing times of a neuron/oscillator. Inset (C) zooms-in on the firing time pattern from (B). Initial conditions are as in Figure 5. Parameters: N=21 , q=4 , τ=0.15 , ϰ=−0.2π , vth=−vr=105 , η¯=2 , δη=6×10−3 , ν=105 correspond to point C in Figure 3. The firing rate was calculated within a sliding time window of δt=5×10−2 .
FIGURE 9
FIGURE 9
Diagram similar to Figure 8 showing a nearly perfect match for asynchronous firing rate (A) and firing times (B) in the QIF network (cyan curves and round markers) and the KS model (black curves and cross markers). Parameters: N=21 , q=4 , τ=0.41 , ϰ=−0.2π , vth=−vr=105 , η¯=2 , δη=6×10−3 , ν=105 correspond to point D in Figure 3. Other notations and settings are as in Figure 8.
FIGURE 10
FIGURE 10
Firing rate (A) and firing times (B) of a three-cluster cyclops state in the QIF network (cyan curves and round markers) and the KS model (black curves and cross markers). (C) Snapshots of the cyclops state phase distributions φn in the KS model at two time instants. The oscillators’ coloring represents their phase. The cyclops states is formed by a solitary oscillator (red) and two coherent clusters, each composed of 10 oscillators (orange and blue). The initial phases are chosen near a cyclops state. Parameters: N=21 , q=2 , τ=0.8 , ϰ=−0.2π , vth=−vr=105 , η¯=2 , δη=0 , ν=105 . Other notations and settings are as in Figure 8.
FIGURE 11
FIGURE 11
Firing rate (A) and firing times (B) in the QIF network demonstrating the transition to full synchronization starting from random initial conditions. Parameters N=1,000 , q=4 , τ=0.15 , ϰ=−0.2π , vth=−vr=100 , η¯=2 , and δη=6×10−3 correspond to point C in Figure 3. The simulations use the algorithm from Pazó and Montbrió (2016), which accounts for the neurons’ refractory time. The mean synaptic activation was calculated within a sliding time window of δt=5×10−2 .
FIGURE 12
FIGURE 12
Firing rate (A) and firing times (B) in the QIF network accompanying the transition from synchronous to asynchronous dynamics. Parameters N=1,000 , q=4 , τ=0.41 , ϰ=−0.2π , vth=−vr=100 , η¯=2 , and δη=6×10−3 correspond to point D in Figure 3. Other notations and settings are as in Figure 11.
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