PaperlayerView exact revision on arXiv

Source-linked AI summary

Nonlinear gap junctions enable long-distance propagation of pulsating calcium waves in astrocyte networks

Mati Goldberg, Maurizio De Pittà, Vladislav Volman, Hugues Berry, Eshel Ben-Jacob

arXiv:1009.2243v1q-bio.NCnlin.CDq-bio.CB

TL;DR

Whether IP3 transfer through astrocyte gap junctions can sustain long-distance regenerative calcium waves, and how intracellular calcium dynamics shape propagation, remains unresolved. A computational model finds that long-distance propagation is favored by nonlinear gap-junction coupling and frequency-modulated calcium oscillations.

  • Problem

    How intracellular calcium dynamics influence propagation distance through the gap-junctional route, and whether that route supports long-distance regenerative waves, remains unclear.

    Source: Introduction · p. 5

  • Method

    The study uses a computational model of a one-dimensional astrocyte network, combining modeled intracellular calcium dynamics with alternative IP3 gap-junction coupling and bifurcation analysis.

    Source: Introduction · p. 6, The ChI model of intracellular Ca2+… · p. 6, Astrocyte coupling · p. 7

  • Results

    Long-distance regenerative calcium-wave propagation is possible with nonlinear gap-junction permeability when most model astrocytes encode incoming IP3 strength through frequency-modulated calcium oscillations.

    Source: Discussion · pp. 19–20

  • Takeaways & Limitations

    In the model, propagation distance and dynamics depend on cells’ encoding properties and their spatial arrangement in composite networks.

    Source: Discussion · p. 22

    The study considers a simplified regular one-dimensional cell chain, enabling bifurcation analysis that would be difficult in higher-dimensional networks.

    Source: Discussion · p. 21

Abstract

from arXiv · show

A new paradigm has recently emerged in brain science whereby communications between glial cells and neuron-glia interactions should be considered together with neurons and their networks to understand higher brain functions. In particular, astrocytes, the main type of glial cells in the cortex, have been shown to communicate with neurons and with each other. They are thought to form a gap-junction-coupled syncytium supporting cell-cell communication via propagating Ca2+ waves. An identified mode of propagation is based on cytoplasm-to-cytoplasm transport of inositol trisphosphate (IP3) through gap junctions that locally trigger Ca2+ pulses via IP3-dependent Ca2+-induced Ca2+ release. It is, however, currently unknown whether this intracellular route is able to support the propagation of long-distance regenerative Ca2+ waves or is restricted to short-distance signaling. Furthermore, the influence of the intracellular signaling dynamics on intercellular propagation remains to be understood. In this work, we propose a model of the gap-junctional route for intercellular Ca2+ wave propagation in astrocytes showing that: (1) long-distance regenerative signaling requires nonlinear coupling in the gap junctions, and (2) even with nonlinear gap junctions, long-distance regenerative signaling is favored when the internal Ca2+ dynamics implements frequency modulation-encoding oscillations with pulsating dynamics, while amplitude modulation-encoding dynamics tends to restrict the propagation range. As a result, spatially heterogeneous molecular properties and/or weak couplings are shown to give rise to rich spatiotemporal dynamics that support complex propagation behaviors. These results shed new light on the mechanisms implicated in the propagation of Ca2+ waves across astrocytes and precise the conditions under which glial cells may participate in information processing in the brain.

Author Summary · Introduction · Methods

The model predicts that long-distance astrocyte Ca2+ waves require nonlinear, thresholded gap-junction coupling and are favored by pulsating frequency-modulated calcium dynamics. It also links propagation range and complex network dynamics to intracellular signaling properties and spatial coupling.

Author Summary

  • Nonlinear, thresholded gap-junction coupling can promote IP3 partial regeneration and enable long-distance regenerative Ca2+ wave propagation.Transferred IP3 must reach a minimum concentration to trigger CICR in a neighboring astrocyte; otherwise propagation ceases.Source: Author Summary · p. 3, Introduction · p. 5
  • Pulsating frequency-modulated Ca2+ oscillations favor long-distance propagation, whereas amplitude-modulated dynamics tend to restrict its range.Pulsating waves have peak widths smaller than their periods; FM, AM, and combined AFM encoding have all been inferred in astrocytes.Source: Author Summary · p. 3, Introduction · p. 5

Introduction

  • With weak coupling, nonlinear gap junctions may help explain complex intracellular oscillations during intercellular waves, while propagation also depends on the cells’ spatial arrangement.The model reports that spatially heterogeneous molecular properties and/or weak couplings can support rich spatiotemporal dynamics.Source: Author Summary · p. 3
  • Astrocytes integrate neuronal inputs and can modulate synaptic activity through gliotransmitter feedback, motivating their consideration alongside neurons in brain computation.Intercellular waves have been reported in cultures and may also occur physiologically, with observed propagation distances of four to 30 astrocytes.Source: Introduction · p. 4
  • Astrocyte Ca2+ oscillation frequency and amplitude vary with cell-specific properties, including intracellular-store Ca2+ content and SERCA pump distribution, density, and activity.This variability motivates examining how intracellular dynamics influence wave propagation.Source: Introduction · p. 5
  • Intercellular propagation can transmit IP3 through gap junctions to trigger CICR, while an alternative route uses extracellular ATP; gap-junction signaling is likely predominant in many astrocyte types.The observed Ca2+ wave is regenerated when transported IP3 reaches the neighboring cell’s threshold for CICR.Source: Introduction · p. 5

The ChI model of intracellular Ca2+ dynamics

The study uses the ChI model to represent intracellular calcium and IP3 signaling, then examines how gap-junction coupling and network setup shape modeled astrocyte signaling.

The ChI model of intracellular Ca2+ dynamics

  • The ChI model extends Li-Rinzel dynamics to couple calcium, IP3-receptor gating, and IP3 production and degradation, and reproduces measured effects of SERCA modulation on oscillation frequency and wave shape.Its pathways include IP3-dependent calcium-induced calcium release, PLCδ-mediated IP3 synthesis, and degradation by 3K and 5P.Source: The ChI model of intracellular Ca2+… · pp. 6–7

Astrocyte coupling

  • The model compares linear, threshold-linear, and sigmoid IP3 exchange between adjacent astrocytes; the latter two transfer IP3 only above a gradient threshold.Sigmoid coupling adds saturation at high gradients, while the threshold-linear form retains a linear flux above threshold. Gap-junction permeability may be actively modulated by signaling factors, motivating tests of regulated coupling.Source: Astrocyte coupling · pp. 7–8

The network model

  • The network is a chain of N astrocytes coupled to their two nearest neighbors, with reflective, absorbing, or periodic boundary conditions.Periodic boundaries make the chain a ring; reflective boundaries have zero flux at the ends, while absorbing ends entrap incoming IP3 flux.Source: The network model · pp. 9–10

Stimulation

  • A fixed-IP3 dummy-cell reservoir stimulates a selected driving cell, usually the first or central cell, and most simulations use constant IP3 input.A square positive-wave stimulus is also applied to the model.Source: Stimulation · p. 10

Results

The results show that long-range Ca2+ wave propagation depends on nonlinear, thresholded gap-junction coupling and on intracellular dynamics that can regenerate pulses. They also identify conditions under which heterogeneous or weakly coupled astrocyte chains support propagation, including complex patterns.

Results

  • Single-cell bifurcation analysis identified AM, FM, and mixed AFM stimulus-encoding regimes, while coupled-cell diagrams showed pulse-like FM oscillations at low IP3 bias.Because oscillations and stable fixed points could coexist, bifurcation diagrams alone could not predict whether stimulation would trigger oscillations, so propagation conditions were tested by simulation.Source: Results · pp. 11–12

Calcium wave propagation along astrocyte chains

  • With FM cells, nonlinear thresholded gap junctions supported Ca2+ pulses across the full chain, whereas linear coupling stopped near one-third of the chain.Nonlinear coupling lets IP3 accumulate enough within a cell to trigger CICR before crossing the junction threshold; this produces local regeneration. Long-range propagation occurred across a broad range of IP3 synthesis and degradation parameters.Source: Calcium wave propagation along… · pp. 12–13
  • AFM chains failed to propagate waves across tested stimulation strengths and coupling types, while FM dynamics regenerated pulses because peak IP3 oscillations remain robust to weaker input.In AFM cells, declining input reduces oscillation peaks and the oscillatory input range is narrower, making propagation more likely to terminate; intracellular properties that switch encoding modes may therefore affect propagation.Source: Calcium wave propagation along… · pp. 14–15
  • In composite chains, higher IP3 production in AFM cells or intervening successive FM cells enabled waves to cross AFM cells that otherwise halted propagation.Successive FM cells increased pulse frequency and IP3 diffusion into the next AFM cell; the authors suggest propagation distance may still be bounded as pulse frequency declines after AFM cells.Source: Calcium wave propagation along… · pp. 16–17

Propagation of complex waves

  • Periodic square-wave stimulation of central cell #21 in 41 coupled FM cells triggered Ca2+ waves along the whole chain, with occasional propagation failures, irregular pulse-like events, and variable pulse shapes.Preliminary time-series analysis suggested deterministic chaos, with positive maximal Lyapunov exponents estimated between 0.020 and 0.050 s-1, depending on the time series.Source: Propagation of complex waves · pp. 17–18

Discussion

The model predicts that long-distance regenerative Ca2+ waves require nonlinear gap-junction permeability and are favored when most astrocytes encode incoming IP3 strength in the frequency of Ca2+ oscillations. It also highlights how cell heterogeneity and spatial organization may shape wave propagation, while noting limits of the simplified 1D network.

Discussion

  • Long-distance regenerative Ca2+ waves require nonlinear gap-junction permeability and are favored when most astrocytes encode incoming IP3 strength in oscillation frequency (FM).The model finds that nonlinear coupling enhances propagation, whereas classic linear coupling leads to fast dissipation; amplitude-modulated encoding tends to restrict propagation range.Source: Discussion · pp. 18–19
  • The model addresses the intracellular IP3-transfer route, expected to be most relevant in neocortex and striatum, where evidence suggests it predominates.Astrocyte wave distances reported experimentally range from 3–4 cells to 30 cells; prior PLCδ-mediated IP3 regeneration models could explain shorter ranges with partial regeneration but omit Ca2+-dependent IP3 degradation, which may hinder propagation.Source: Discussion · p. 19
  • Heterogeneity in astrocyte responses may strongly affect intercellular wave propagation, including through mixed amplitude- and frequency-modulated (AFM-FM) networks.The authors suggest variability in PLCδ or SERCA parameters as possible sources; calreticulin may dynamically shift SERCA Ca2+ affinity from 170 to ~400 nM.Source: Discussion · p. 20
  • Nonlinear gap-junction coupling is a permissive factor for long-range signaling, but its existence in astrocytes has not been directly demonstrated.Dynamic gap-junction resistance is known in cardiac and other cells, and PKC activation can block astroglial gap-junction communication and inhibit Ca2+ wave spread; the model therefore uses a generic nonlinearity to explore its qualitative effects.Source: Discussion · pp. 20–21
  • The regular 1D chain simplifies quasi-2D astrocyte networks, so whether linear gap junctions remain limited to local propagation in higher-dimensional networks is unresolved.The model also predicts that spatially heterogeneous gap-junction permeabilities and reduced coupling strength can produce rich dynamics, possibly including spatiotemporal chaos; experiments targeting gap-junction regulation could test whether networks self-regulate this complexity.Source: Discussion · pp. 21–22
  • In composite networks, wave propagation dynamics and distance depend on both cells’ encoding properties and their spatial arrangement, suggesting a possible role for astrocyte contact spacing.The authors propose that orderly spacing of neighboring astrocyte cell bodies may influence wave propagation and the computational properties of astrocyte networks.Source: Discussion · p. 22
  • The authors frame astrocyte Ca2+ waves as a possible route for long-range, delayed regulation of neuronal and synaptic activity, motivating future joint experimental and modeling studies.Astrocytes and neurons exchange molecular signals and operate as interacting networks on distinct timescales; how their mutual regulation affects neuronal-network dynamics in health and disease remains an open question.Source: Discussion · p. 22

Figure 2 · Figure 3 · Nonlinear gap junctions enable long-distance propagation of pulsating

The supplied passages identify the authors but do not provide findings for the merged sections.

calcium waves in astrocyte networks

SUPPLEMENTARY INFORMATION · 1. The ChI model for intracellular Ca2+ dynamics

The ChI model extends the Li–Rinzel description of astrocytic Ca2+ dynamics by making IP3 a third, time-varying variable coupled to cytoplasmic Ca2+. It simplifies IP3 signaling into tractable production and degradation processes while allowing an external IP3 input.

1. The ChI model for intracellular Ca2+ dynamics

  • Astrocytic cytoplasmic Ca2+ reflects SERCA pumping into the ER, passive ER leak, and IP3R-mediated release, whose Ca2+-dependent feedback can terminate a release surge.Small-to-intermediate Ca2+ levels promote IP3R activity, while large concentrations inhibit it; the three fluxes are classically modeled by Li and Rinzel.Source: §1 · p. 38
  • Unlike the Li–Rinzel model, which treats IP3 as constant, the ChI model accounts for IP3 formation and degradation regulated by cytoplasmic Ca2+.The signaling pathway therefore couples IP3 concentration to Ca2+ dynamics.Source: §1 · p. 39
  • The ChI model extends Li–Rinzel by adding cytoplasmic IP3 as a third dynamical variable alongside Ca2+ concentration and the fraction of inactive IP3R channels.The classical Li–Rinzel formulations for the Ca2+ fluxes and IP3R gating are retained.Source: §1 · p. 39
  • Endogenous IP3 is produced from PIP2 by PLCδ and degraded by IP3-3K phosphorylation or IP-5P dephosphorylation.PLCδ activation by Ca2+ is cooperative, with a Hill exponent close to 2, and PLCδ is inhibited at IP3 concentrations above circa 1 µM.Source: §1 · p. 39
  • The ChI model represents PLCδ production with Ca2+-dependent Hill regulation and IP3 inhibition, and approximates IP-5P degradation as linear because its Michaelis constant exceeds physiological IP3 levels.The IP3-3K regulation is lumped into a product of a fourth-order Ca2+ Hill function and a first-order IP3 Hill function.Source: §1 · pp. 39–40
  • In the ChI model, IP3 concentration is a third dynamical variable; vbias defines external IP3 input, for example from a neighboring astrocyte coupled through a gap junction, and vbias=0 yields equation (3) from equation (S4).Source: §1 · p. 40

2. Bifurcation analysis of the ChI model and encoding modes

The ChI model links IP3 bias to bifurcation-dependent Ca2+ encoding in isolated cells, while coupled-chain dynamics show that IP3-triggered CICR can propagate pulsating Ca2+ waves when neighbor-directed IP3 pulses remain effective.

a) Isolated cell

  • As IP3 bias rises, isolated ChI cells progress from a low steady state through oscillations to a high steady state, with oscillation onset occurring through Hopf or SNIC bifurcation depending on cell parameters.The overall scenario persists across realistic parameter ranges.Source: a) Isolated cell · p. 41
  • Across the oscillatory range, AM changes Ca2+ amplitude by more than twofold with nearly constant frequency, FM changes frequency by more than twofold with nearly constant amplitude, and AFM changes both by more than twofold.Source: a) Isolated cell · pp. 41–42
  • FM oscillations are usually pulse-like, whereas AM oscillations are near-sinusoidal; as IP3 bias rises, AFM amplitude increases and period decreases, while FM Ca2+ amplitude stays nearly constant as period varies markedly.The oscillatory IP3-bias range is narrower in AFM than FM conditions, and SERCA transport is larger under FM conditions.Source: a) Isolated cell · p. 42

b) Coupled cell network

  • In coupled cells, the stimulated central cell retains SNIC onset near 0.72 µM IP3 bias, but higher bias brings period-doubling cascades and coexisting stable oscillatory regimes, producing complex dynamics.FM pulse-like oscillations at low IP3 can become complex at higher bias; bifurcation diagrams are numerically arduous and cannot predict whether neighbor input will trigger oscillations, prompting simulations.Source: b) Coupled cell network · pp. 42–43
  • A sufficiently large IP3 pulse triggers CICR and a Ca2+ pulse, followed by an IP3 pulse that diffuses to neighboring cells and sustains pulsating-wave propagation; an undersized pulse fails to trigger the next cell and stops the wave.Propagation distance depends strongly on gap-junction coupling and cell type (AFM or FM).Source: b) Coupled cell network · pp. 43–44

Parameter Value

This section lists the model’s Li-Rinzel core, IP3 metabolism, gap-junction coupling, and nonlinear gap-junction parameter values.

Li-Rinzel core parameters

  • The Li-Rinzel core uses C0 2.0 μM, c1 0.185, and rC 6.0 s^-1, with rL 0.11 s^-1, vER 0.9 μM·s^-1, and d1 0.13 μM.These are listed as total cell free [Ca2+] per cytosolic volume, ER-to-cytosol volume ratio, maximal CICR rate, maximal ER Ca2+ leak, maximal SERCA uptake, and IP3 dissociation constant, respectively.Source: Parameter Value · p. 46
  • The core dissociation constants are d2 1.049 μM, d3 0.9434 μM, and d5 0.08234 μM; a2 is 0.200 s^-1 and KER is 0.10 / 0.05 μM.d2 is the Ca2+ inactivation constant, d3 the IP3 constant, d5 the Ca2+ activation constant, a2 the IP3R Ca2+-inhibition binding rate, and KER the SERCA Ca2+ affinity.Source: Parameter Value · p. 46

IP3 metabolism parameters

  • IP3 metabolism parameters include vδ 0.12 / 0.7 μM·s^-1, KPLCδ 0.1 μM, v3K 4.5 s^-1, K3K 0.7 μM, K3 1 μM, and r5P 0.04 / 0.21 s^-1.The listed quantities specify PLCδ synthesis and Ca2+ affinity, IP3 degradation by 3K and IP-5P, and the associated half-maximal or half-saturation constants.Source: Parameter Value · p. 46

Gap-junction parameters

Nonlinear gap-junction parameters

  • Nonlinear gap-junction parameters are shift 0.3 μM, scale 0.05 μM, and bias 0.8 / 1.0 μM.Shift is the half-maximal diffusion IP3 threshold, scale the IP3 slope factor, and bias the imposed IP3 level in the stimulated cell.Source: Parameter Value · p. 46

Nonlinear gap junctions enable long-distance propagation of pulsating

The paper proposes that long-distance regenerative Ca2+ wave propagation in astrocyte networks requires nonlinear gap-junction coupling. It further argues that pulsating, frequency-modulated Ca2+ dynamics favor propagation, while amplitude-modulated dynamics tend to limit its range.

calcium waves in astrocyte networks

Supplementary Figures - Captions

The supplementary figures detail how astrocyte calcium dynamics depend on encoding regime and how gap-junctional IP3 transfer can support, limit, or shape wave propagation. They also show complex oscillatory behavior and broad interpulse-interval distributions in some cells.

Figure SI1

  • FM oscillations arise through a SNIC bifurcation, producing arbitrarily small frequencies at nearly constant amplitude, whereas AFM amplitude and period are controlled by supercritical Hopf and saddle-node limit-cycle bifurcations.Source: Supplementary Figures - Captions · p. 47, Figure SI1

Figure SI2

  • In the seven-cell FM chain, oscillations become highly complex above an IP3 bias of about 0.8 µM; above 1.1 µM, their amplitude shrinks and multiple complex stable orbits coexist.A rapid cascade of period-doubling bifurcations produces limit cycles with numerous folds that could not be precisely rendered.Source: Supplementary Figures - Captions · p. 47, Figure SI2

Figure SI3

  • Suprathreshold IP3 transfer through sigmoid gap junctions triggers regenerative calcium pulses in successive FM cells, but propagation stops when influx falls below the level needed for CICR.In the illustrated five-cell chain, pulses reach A2 but not A4 or A5.Source: Supplementary Figures - Captions · p. 48, Figure SI3

Figure SI4

  • Propagation patterns are illustrated for a 12-cell FM chain with sigmoid-like gap junctions under periodic or absorbing boundary conditions and central- or end-cell stimulation.Source: Supplementary Figures - Captions · p. 48, Figure SI4

Figure SI5

Figure SI6

  • Interpulse-interval distributions are broad in several simulated cells, with large intervals occurring as commonly as smaller ones.The distributions cover cells A1–A10 and A11–A20; each panel indicates the number of pulses counted.Source: Supplementary Figures - Captions · p. 48, Figure SI6
Loading 1009.2243v1…