Purkinje neuron network (Zang et al. 2020)

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Both spike rate and timing can transmit information in the brain. Phase response curves (PRCs) quantify how a neuron transforms input to output by spike timing. PRCs exhibit strong firing-rate adaptation, but its mechanism and relevance for network output are poorly understood. Using our Purkinje cell (PC) model we demonstrate that the rate adaptation is caused by rate-dependent subthreshold membrane potentials efficiently regulating the activation of Na+ channels. Then we use a realistic PC network model to examine how rate-dependent responses synchronize spikes in the scenario of reciprocal inhibition-caused high-frequency oscillations. The changes in PRC cause oscillations and spike correlations only at high firing rates. The causal role of the PRC is confirmed using a simpler coupled oscillator network model. This mechanism enables transient oscillations between fast-spiking neurons that thereby form PC assemblies. Our work demonstrates that rate adaptation of PRCs can spatio-temporally organize the PC input to cerebellar nuclei.
1 . Zang Y, Hong S, De Schutter E (2020) Firing rate-dependent phase responses of Purkinje cells support transient oscillations. Elife [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Neuron or other electrically excitable cell; Realistic Network;
Brain Region(s)/Organism: Cerebellum;
Cell Type(s): Cerebellum Purkinje GABA cell;
Gap Junctions:
Simulation Environment: NEURON; MATLAB;
Model Concept(s): Phase Response Curves; Action Potentials; Spatio-temporal Activity Patterns; Synchronization; Action Potential Initiation; Oscillations;
Implementer(s): Zang, Yunliang ; Hong, Sungho [shhong at oist.jp];
Search NeuronDB for information about:  Cerebellum Purkinje GABA cell;
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    Sinusoidal + Fluctuating current model for temporally-modulated synaptic bombardment

  The present version implements and generate a realization of an Ornstein-Uhlenbeck (OU) process
  (see Cox & Miller, 1969; see Tuckwell) to mimick the somatic impact of linearly adding EPSPs and
  IPSPs. Thus, it generates and injects in the specified neuronal compartment a fluctuating current
  waveform (a noise), characterized by a gauss-distributed amplitude, where neighboring amplitude 
  samples are by definition linearly correlated on a time scale set by the correlation time-length
  "tau" of the process.
  The numerical scheme for integration of OU processes takes advantage of the fact that these
  processes are gaussian, which led to an exact update rule independent of the time step dt..
  (see Gillespie DT, Am J Phys 64: 225, 1996):

  x(t+dt) = x(t) + (1. - exp(-dt/tau)) * (m - x) + sqrt(1.-exp(-2.*dt/tau)) * s * N(0,1)  
  where N(0,1) is a normal random number (avg=0, sigma=1)..

  Please note that only fixed integration time-step methods makes sense, since the stochastic current
  synthesized by the present mechanism is produced randomly and on-line. In other words, it is wrong to
  assume that neglecting the present integration step, reducing it and resynthesizing the current, lead
  to the same overall trajectory in the compartment output voltage.

  As opposed to the previously developed mechanisms Ifluct1.mod (see the ModelDB), here the MEAN of the
  current is sinusoidally oscillating as indicated below, as a function of time: A * sin (2 pi f t ).


  This mechanism is implemented as a nonspecific current defined as a point process, mimicking a current-
  clamp stimulation protocol, injecting a sinusoidally oscillating waveform overlapped to a noisy component.
  I(t)  = m +  A * sin (2 pi f t) + x(t)

  Since this is an electrode current, positive values of i depolarize the cell and in the presence of the
  extracellular mechanism there will be a change in vext since i is not a transmembrane current but a current
  injected directly to the inside of the cell.

 Koendgen, H., Geisler, C., Fusi, S., Wang, X.-J., Luescher, H.-R., and Giugliano, M. (2007). 
 Arsiero, M., Luescher, H.-R., Lundstrom, B.N., and Giugliano, M. (2007). 
 La Camera, G., Rauch, A., Thurbon, D., Luescher, H.-R., Senn, W., and Fusi, S. (2006). 
 Giugliano, M., Darbon, P., Arsiero, M., Luescher, H.-R., and Streit, J. (2004). 
 Rauch, A., La Camera, G., Luescher, H.-R., Senn, W., and Fusi, S. (2003). 
 Boucsein, C. Tetzlaff, T., Meier, R., Aertsen, A., and B. Naundorf (2009)

 The present mechanism has been inspired by "Gfluct.mod", by A. Destexhe (1999), as taken from ModelDB.
 Destexhe, A., Rudolph, M., Fellous, J-M. and Sejnowski, T.J. (2001). 

 M. Giugliano, Theoretical Neurobiology, Department of Biomedical Sciences, University of Antwerp, Antwerp
 		and Brain Mind Institute, EPFL Lausanne
 V. Delattre, Brain Mind Institute, EPFL Lausanne


  The mechanism takes as input the following parameters (reported with their default values):

    m   = 0. (nA)       : DC offset of the overall current
    s   = 0. (nA)       : square root of the steady-state variance of the (noisy) stimulus component
    tau = 2. (ms)       : steady-state correlation time-length of the (noisy) stimulus component
    amp = 0. (nA)       : amplitude of the (sinusoidal) stimulus component
    freq= 0. (Hz)       : steady-state correlation time-length of the (noisy) stimulus component



    POINT_PROCESS Isinunoisy
    RANGE amp, i, freq, m, s, tau, x, new_seed,onset,np

    (nA) = (nanoamp)

    dt   (ms)
    m   = 0. (nA)       : DC offset of the overall current
    s   = 0. (nA)       : square root of the steady-state variance of the (noisy) stimulus component
    tau = 2. (ms)       : steady-state correlation time-length of the (noisy) stimulus component
    amp = 0. (nA)       : amplitude of the (sinusoidal) stimulus component
    freq= 0. (Hz)       : steady-state correlation time-length of the (noisy) stimulus component
    phas= 0. (HZ)       : phase of the sinusoidal input current
    fr2 = 5.(Hz)       : frequency of the sinusoidal input current
    onset = 10
    np = 100000000000   : the number of sinusoidal current waves exerted.

    i (nA)              : overall sinusoidal noisy current
    x                   : state variable

    i = m
    x = m               : to reduce the transient, the state is set to its (expected) steady-state    

    SOLVE oup
    if (tau <= 0) {  x = m + s  * normrand(0,1) }  : remember: true "white-noise" is impossible to generate anyway..
    if ((t<onset)||(t>onset+(1000/freq)*np)) {
        i = x
    } else {
        i = x + amp * sin(0.0062831853071795866 * freq * t)
:    i = x + amp * sin(0.0062831853071795866 * freq * t)

PROCEDURE oup() {       : uses "Scop" function normrand(mean, std_dev)
if (tau > 0) {  x = x + (1. - exp(-dt/tau)) * (m - x) + sqrt(1.-exp(-2.*dt/tau)) * s  * normrand(0,1)}

PROCEDURE new_seed(seed) {      : procedure to set the seed


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