A network of AOB mitral cells that produces infra-slow bursting (Zylbertal et al. 2017)

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Infra-slow rhythmic neuronal activity with very long (> 10 s) period duration was described in many brain areas but little is known about the role of this activity and the mechanisms that produce it. Here we combine experimental and computational methods to show that synchronous infra-slow bursting activity in mitral cells of the mouse accessory olfactory bulb (AOB) emerges from interplay between intracellular dynamics and network connectivity. In this novel mechanism, slow intracellular Na+ dynamics endow AOB mitral cells with a weak tendency to burst, which is further enhanced and stabilized by chemical and electrical synapses between them. Combined with the unique topology of the AOB network, infra-slow bursting enables integration and binding of multiple chemosensory stimuli over prolonged time scale. The example protocol simulates a two-glomeruli network with a single shared cell. Although each glomerulus is stimulated at a different time point, the activity of the entire population becomes synchronous (see paper Fig. 8)
1 . Zylbertal A, Yarom Y, Wagner S (2017) Synchronous Infra-Slow Bursting in the Mouse Accessory Olfactory Bulb Emerge from Interplay between Intrinsic Neuronal Dynamics and Network Connectivity. J Neurosci 37:2656-2672 [PubMed]
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Model Information (Click on a link to find other models with that property)
Model Type: Realistic Network;
Brain Region(s)/Organism: Olfactory bulb;
Cell Type(s): Olfactory bulb (accessory) mitral cell;
Channel(s): I CAN; Na/Ca exchanger; Na/K pump; I Calcium; I Na,t;
Gap Junctions: Gap junctions;
Simulation Environment: NEURON; Python;
Model Concept(s): Bursting; Synchronization; Activity Patterns; Oscillations; Persistent activity; Olfaction;
Implementer(s): Zylbertal, Asaph [asaph.zylbertal at mail.huji.ac.il];
Search NeuronDB for information about:  I Na,t; I CAN; I Calcium; Na/Ca exchanger; Na/K pump;
: Calcium ion accumulation with radial diffusion, buffering and pumping

    SUFFIX cadp
    USEION ca READ cao, cai, ica WRITE cai, ica
    RANGE ica_pmp, ex_buffer_ratio, f, AvgCaExBuffer, diam_factor, TotalPump, ca, CaEndBuffer, EndBuffer, CaExBuffer, ExBuffer, pump, pumpca
    GLOBAL vrat, TotalEndBuffer, TotalExBuffer, k1, k2, k3, k4, k1bufend, k2bufend, k1bufex, k2bufex, DCa, fl_ratio, dep_factor
    USEION dep READ depi VALENCE 1

    NONSPECIFIC_CURRENT icont                                                       

DEFINE Nannuli 4

    (molar) = (1/liter)
    (mM) = (millimolar)
    (um) = (micron)
    (mA) = (milliamp)
    FARADAY = (faraday) (10000 coulomb)
    PI = (pi) (1)
    (mol) = (1)



    DCa = 0.6 (um2/ms)
    k1bufend = 100 (/mM-ms) : Yamada et al. 1989
    k2bufend = 0.1 (/ms)
    TotalEndBuffer = 0.003 (mM)

    k1bufex = 100 (/mM-ms) 
    k2bufex = 0.017 (/ms)       :   Based on OGB-1 kd
    TotalExBuffer = 0.04 (mM)   : 

    k1 = 1 (/mM-ms)
    k3 = 1 (/ms)
                            : to eliminate pump, set TotalPump to 0 in hoc
    TotalPump = 1e-14 (mol/cm2)

    fl_ratio=14 (1)

    diam_factor=1 (1)
    dep_factor=1 (1)


    diam (um)
    L (um)
    ica (mA/cm2)
    cai (mM)
    vrat[Nannuli] : numeric value of vrat[i] equals the volume
                 : of annulus i of a 1um diameter cylinder
                 : multiply by diam^2 to get volume per um length
    k2          (/ms)
    k4          (/mM-ms)

    ka_end (/mM)
    ka_ex (/mM)

    B0end (mM)
    B0ex (mM)

    cao (mM)
    ica_pmp (mA/cm2)
    icont (mA/cm2)
    parea (um)
    AvgCaExBuffer (mM)  : [CaExBuffer] averaged over all shells.

    f    (1)
    diamf   (um)


CONSTANT { volo = 1e10 (um2) }

    : ca[0] is equivalent to cai
    : ca[] are very small, so specify absolute tolerance
    ca[Nannuli] (mM) <1e-10>
    CaEndBuffer[Nannuli] (mM)
    EndBuffer[Nannuli] (mM)
    CaExBuffer[Nannuli] (mM)
    ExBuffer[Nannuli] (mM)

    pump (mol/cm2)
    pumpca (mol/cm2)



    SOLVE state METHOD sparse

    ica = ica_pmp
    icont = -ica_pmp
    AvgCaExBuffer = 0.0
    FROM i=0 TO Nannuli-1 {
        AvgCaExBuffer = AvgCaExBuffer + (CaExBuffer[i] * vrat[i])

    AvgCaExBuffer = AvgCaExBuffer * (4/PI)
    f = TotalExBuffer + (fl_ratio - 1) * AvgCaExBuffer

LOCAL factors_done

    k2=sqrt(cai/cao)    :Set the equilibrium at cai0_ca_ion

    parea = PI*diamf
    pump = TotalPump/(1 + (cai*k1/k2))
    pumpca = TotalPump - pump
    if (factors_done == 0) {    : flag becomes 1 in the first segment
        factors_done = 1        : all subsequent segments will have
        factors()               : vrat = 0 unless vrat is GLOBAL

    ka_end = k1bufend/k2bufend
    ka_ex = k1bufex/k2bufex

    B0end = TotalEndBuffer/(1 + ka_end*cai)
    B0ex = TotalExBuffer/(1 + ka_ex*cai)

:    ex_buffer_ratio = 0.0
    FROM i=0 TO Nannuli-1 {
        ca[i] = cai
        EndBuffer[i] = B0end
        CaEndBuffer[i] = TotalEndBuffer - B0end
        ExBuffer[i] = B0ex
        CaExBuffer[i] = TotalExBuffer - B0ex
        :ex_buffer_ratio = ex_buffer_ratio + (CaExBuffer[i] * vrat[i])

    :ex_buffer_ratio=(ex_buffer_ratio/(PI/4))/(TotalExBuffer - (ex_buffer_ratio/(PI/4)))


LOCAL frat[Nannuli]     : scales the rate constants for model geometry

PROCEDURE factors() {
    LOCAL r, dr2
    r = 1/2                 : starts at edge (half diam)
    dr2 = r/(Nannuli-1)/2   : full thickness of outermost annulus,
                            : half thickness of all other annuli
    vrat[0] = 0
    frat[0] = 2*r
    FROM i=0 TO Nannuli-2 {
        vrat[i] = vrat[i] + PI*(r-dr2/2)*2*dr2  : interior half
        r = r - dr2
        frat[i+1] = 2*PI*r/(2*dr2)              : outer radius of annulus
                                                : div by distance between centers
        r = r - dr2
        vrat[i+1] = PI*(r+dr2/2)*2*dr2 : outer half of annulus

LOCAL dsq, dsqvol   : can't define local variable in KINETIC block
                    : or use in COMPARTMENT statement

KINETIC state {
    COMPARTMENT i, diamf*diamf*vrat[i] {ca CaIndBuffer IndBuffer CaExBuffer ExBuffer}
    COMPARTMENT (1e10)*parea {pump pumpca}
    COMPARTMENT volo {cao}

    :LONGITUDINAL_DIFFUSION i, DCa*diamf*diamf*vrat[i] {ca}

    ~ ca[0] + pump <-> pumpca (k1*(1-depi*dep_factor)*parea*(1e10), k2*(1-depi*dep_factor)*parea*(1e10))
    ~ pumpca <-> pump + cao (k3*(1-depi*dep_factor)*parea*(1e10), k4*(1-depi*dep_factor)*parea*(1e10))

    CONSERVE pump + pumpca = TotalPump * parea * (1e10)
    ica_pmp = 2*FARADAY*(f_flux - b_flux)/parea

    : all currents except pump
    ~ ca[0] << (-(ica-ica_pmp)*PI*diamf/(2*FARADAY)) : ica is Ca efflux

    FROM i=0 TO Nannuli-2 {
        ~ ca[i] <-> ca[i+1] (DCa*frat[i+1], DCa*frat[i+1])
    dsq = diamf*diamf
    FROM i=0 TO Nannuli-1 {
        dsqvol = dsq*vrat[i]
        ~ ca[i] + EndBuffer[i] <-> CaEndBuffer[i] (k1bufend*dsqvol, k2bufend*dsqvol)
        ~ ca[i] + ExBuffer[i] <-> CaExBuffer[i] (k1bufex*dsqvol, k2bufex*dsqvol)

    cai = ca[0]