brainpy.dyn.neurons.PinskyRinzelModel#

class brainpy.dyn.neurons.PinskyRinzelModel(size, keep_size=False, gNa=30.0, gK=15.0, gCa=10.0, gAHP=0.8, gC=15.0, gL=0.1, ENa=60.0, EK=- 75.0, ECa=80.0, EL=- 60.0, gc=2.1, V_th=20.0, Cm=3.0, p=0.5, A=1.0, Vs_initializer=OneInit(value=- 64.6), Vd_initializer=OneInit(value=- 64.5), Ca_initializer=OneInit(value=0.2), noise=None, method='exp_auto', name=None, mode=NormalMode)[source]#

The Pinsky and Rinsel (1994) model.

The Pinsky and Rinsel (1994) model 7 is a 2-compartment (soma and dendrite), conductance-based (Hodgin-Huxley type) model of a hippocampal CA3 pyramidal neuron. It is a reduced version of an earlier, 19-compartment model by Traub, et. al. (1991) 8. This model demonstrates how similar qualitative and quantitative spiking behaviors can be obtained despite the reduction in model complexity.

Specifically, this model demonstrates calcium bursting behavior and how the ‘ping-pong’ interplay between somatic and dendritic currents results in a complex shape of the burst.

../../../../_images/Pinsky-Rinzel-model-illustration.png

Mathematically, the model is given by:

\[\begin{split} \begin{aligned} &\mathrm{C}_{\mathrm{m}} \mathrm{V}_{\mathrm{s}}^{\prime}=-\mathrm{I}_{\mathrm{Leak}}-\mathrm{I}_{\mathrm{Na}}-\mathrm{I}_{\mathrm{K}_{\mathrm{DR}}}-\frac{\mathrm{I}_{\mathrm{DS}}}{\mathrm{p}}+\frac{\mathrm{I}_{\mathrm{S}_{\mathrm{app}}}}{\mathrm{p}} \\ &\mathrm{C}_{\mathrm{m}} \mathrm{V}_{\mathrm{d}}^{\prime}=-\mathrm{I}_{\mathrm{Leak}}-\mathrm{I}_{\mathrm{Ca}}-\mathrm{I}_{\mathrm{K}_{\mathrm{Ca}}}-\mathrm{I}_{\mathrm{K}_{\mathrm{AHP}}}+\frac{\mathrm{I}_{\mathrm{SD}}}{(1-\mathrm{p})}+\frac{\mathrm{I}_{\mathrm{D}_{\mathrm{app}}}}{(1-\mathrm{p})} \\ &\frac{\mathrm{dCa}}{\mathrm{dt}}=-0.13 \mathrm{I}_{\mathrm{Ca}}-0.075 \mathrm{Ca} \end{aligned}\end{split}\]

The currents of the model are functions of potentials as follows:

\[\begin{split}\begin{aligned} \mathrm{I}_{\mathrm{Na}} &=\mathrm{g}_{\mathrm{Na}} m_{\infty}^{2}\left(\mathrm{~V}_{\mathrm{s}}\right) h\left(\mathrm{~V}_{\mathrm{s}}-\mathrm{V}_{\mathrm{Na}}\right) \\ \mathrm{I}_{\mathrm{K}_{\mathrm{DR}}} &=\mathrm{g}_{\mathrm{K}_{\mathrm{DR}}} n\left(\mathrm{~V}_{\mathrm{s}}-\mathrm{V}_{\mathrm{K}}\right) \\ \mathrm{I}_{\mathrm{Ca}} &=\mathrm{g}_{\mathrm{Ca}}{ }^{2}\left(\mathrm{~V}_{\mathrm{d}}-\mathrm{V}_{\mathrm{N}}\right) \\ \mathrm{I}_{\mathrm{K}_{\mathrm{Ca}}} &=\mathrm{g}_{\mathrm{k}_{\mathrm{Ca}}} C \chi(\mathrm{Ca})\left(\mathrm{V}_{\mathrm{d}}-\mathrm{V}_{\mathrm{Ca}}\right) \\ \mathrm{I}_{\mathrm{K}_{\mathrm{AHP}}} &=\mathrm{g}_{\mathrm{K}_{\mathrm{AHP}}} q\left(\mathrm{~V}_{\mathrm{d}}-\mathrm{V}_{\mathrm{K}}\right) \\ \mathrm{I}_{\mathrm{SD}} &=-\mathrm{I}_{\mathrm{DS}}=\mathrm{g}_{\mathrm{c}}\left(\mathrm{V}_{\mathrm{d}}-\mathrm{V}_{\mathrm{s}}\right) \\ \mathrm{I}_{\mathrm{Leak}} &=\mathrm{g}_{\mathrm{L}}\left(\mathrm{V}-\mathrm{V}_{\mathrm{L}}\right) \end{aligned}\end{split}\]

The activation and inactivation variables should satisfy these equations

\[\begin{split} \begin{aligned} \omega^{\prime}(\mathrm{V}) &=\frac{\omega_{\infty}(\mathrm{V})-\omega}{\tau_{\omega}(\mathrm{V})} \\ \omega_{\infty}(\mathrm{V}) &=\frac{\alpha_{\omega}(\mathrm{V})}{\alpha_{\omega}(\mathrm{V})+\beta_{\omega}(\mathrm{V})} \\ \tau_{\omega}(\mathrm{V}) &=\frac{1}{\alpha_{\omega}(\mathrm{V})+\beta_{\omega}(\mathrm{V})} \end{aligned}\end{split}\]

where, independently, we consider \(\omega = h, n, s, m, c, q\).

The rate functions are defined as follows

\[\begin{split} \begin{aligned} \alpha_{m}\left(\mathrm{~V}_{\mathrm{s}}\right) &=\frac{0.32\left(-46.9-\mathrm{V}_{\mathrm{s}}\right)}{\exp \left(\frac{-46.9-\mathrm{V}_{\mathrm{s}}}{4}\right)-1} \\ \beta_{m}\left(\mathrm{~V}_{\mathrm{s}}\right) &=\frac{0.28\left(\mathrm{~V}_{\mathrm{s}}+19.9\right)}{\exp \left(\frac{\mathrm{V}_{\mathrm{s}}+19.9}{5}\right)-1}, \\ \alpha_{n}\left(\mathrm{~V}_{\mathrm{s}}\right) &=\frac{0.016\left(-24.9-\mathrm{V}_{\mathrm{s}}\right)}{\exp \left(\frac{-24.9-\mathrm{V}_{\mathrm{s}}}{5}\right)-1} \\ \beta_{n}\left(\mathrm{~V}_{\mathrm{s}}\right) &=0.25 \exp \left(-1-0.025 \mathrm{~V}_{\mathrm{s}}\right) \\ \alpha_{h}\left(\mathrm{~V}_{\mathrm{s}}\right) &=0.128 \exp \left(\frac{-43-\mathrm{V}_{\mathrm{s}}}{18}\right) \\ \beta_{h}\left(\mathrm{~V}_{\mathrm{s}}\right) &=\frac{4}{1+\exp \left(\frac{\left(-20-\mathrm{V}_{\mathrm{s}}\right.}{5}\right)}, \\ \alpha_{s}\left(\mathrm{~V}_{\mathrm{d}}\right) &=\frac{1.6}{1+\exp \left(-0.072\left(\mathrm{~V}_{\mathrm{d}}-5\right)\right)} \\ \beta_{s}\left(\mathrm{~V}_{\mathrm{d}}\right) &=\frac{0.02\left(\mathrm{~V}_{\mathrm{d}}+8.9\right)}{\exp \left(\frac{\left(\mathrm{V}_{\mathrm{d}}+8.9\right)}{5}\right)-1}, \\ \alpha_{C}\left(\mathrm{~V}_{\mathrm{d}}\right) &=\frac{\left(1-H\left(\mathrm{~V}_{\mathrm{d}}+10\right)\right) \exp \left(\frac{\left(\mathrm{V}_{\mathrm{d}}+50\right)}{11}-\frac{\left(\mathrm{V}_{\mathrm{d}}+53.5\right)}{27}\right)}{18.975}+H\left(\mathrm{~V}_{\mathrm{d}}+10\right)\left(2 \exp \left(\frac{\left(-53.5-\mathrm{V}_{\mathrm{d}}\right.}{27}\right)\right) \\ \beta_{C}\left(\mathrm{~V}_{\mathrm{d}}\right) &=\left(1-H\left(\mathrm{~V}_{\mathrm{d}}+10\right)\right)\left(2 \exp \left(\frac{\left(-53.5-\mathrm{V}_{\mathrm{d}}\right)}{27}\right)-\alpha_{c}\left(\mathrm{~V}_{\mathrm{d}}\right)\right) \\ \alpha_{q}(\mathrm{Ca}) &=\min (0.00002 \mathrm{Ca}, 0.01) \\ \beta_{q}(\mathrm{Ca}) &=0.001 \\ \chi(\mathrm{Ca}) &=\min \left(\frac{\mathrm{Ca}}{250}, 1\right) \end{aligned}\end{split}\]

The standard values of the parameters are given below. The maximal conductances (in \(\mathrm{mS} / \mathrm{cm}^{2}\)) are \(\bar{g}_{L}=0.1\), \(\bar{g}_{\mathrm{Na}}=30\), \(\bar{g}_{\mathrm{K}-\mathrm{DR}}=15\), \(\bar{g}_{\mathrm{Ca}}=10\), \(\bar{g}_{\mathrm{K}-\mathrm{AHP}}=0.8\), \(\bar{g}_{\mathrm{K}-\mathrm{C}}=15\), \(\bar{g}_{\mathrm{NMDA}}=0.0\) and \(\bar{g}_{\mathrm{AMPA}}=0.0\). The reversal potentials (in \(\mathrm{mV}\) ) are \(V_{\mathrm{Na}}=120, V_{\mathrm{C}}=140, V_{\mathrm{K}}=-15 \mathrm{mV})\) are \(V_{\mathrm{Na}}=120, V_{\mathrm{Ca}}=140, V_{\mathrm{K}}=-15, $V_{L}=0\) and \(V_{\text {Syn }}=60\). The applied currents (in \(\mu \mathrm{A} / \mathrm{cm}^{2}\) ) are \(I_{s}=-0.5\) and \(I_{d}=0.0\). The coupling parameters are \(g_{c}=2.1 \mathrm{mS} / \mathrm{cm}^{2}\) and \(p=0.5\). The capacitance, \(C_{M}\), is \(3 \mu \mathrm{F} / \mathrm{cm}^{2}\) and \(\chi(C a)=\min (C a / 250,1)\). Values for these parameters, and these function definitions, are taken from Traub et al, 1991.

Parameters

References

7

Pinsky, Paul F., and John Rinzel. “Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons.” Journal of computational neuroscience 1.1 (1994): 39-60.

8

Traub, R. D., Wong, R. K., Miles, R., & Michelson, H. (1991). A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. Journal of neurophysiology, 66(2), 635-650.

__init__(size, keep_size=False, gNa=30.0, gK=15.0, gCa=10.0, gAHP=0.8, gC=15.0, gL=0.1, ENa=60.0, EK=- 75.0, ECa=80.0, EL=- 60.0, gc=2.1, V_th=20.0, Cm=3.0, p=0.5, A=1.0, Vs_initializer=OneInit(value=- 64.6), Vd_initializer=OneInit(value=- 64.5), Ca_initializer=OneInit(value=0.2), noise=None, method='exp_auto', name=None, mode=NormalMode)[source]#

Methods

__init__(size[, keep_size, gNa, gK, gCa, ...])

alpha_c(Vd)

alpha_h(Vs)

alpha_m(Vs)

alpha_n(Vs)

alpha_q(Ca)

alpha_s(Vd)

beta_c(Vd)

beta_h(Vs)

beta_m(Vs)

beta_n(Vs)

beta_q(Ca)

beta_s(Vd)

clear_input()

Function to clear inputs in the neuron group.

dCa(Ca, t, s, Vd)

dVd(Vd, t, s, q, c, Ca, Vs)

dVs(Vs, t, h, n, Vd)

dc(c, t, Vd)

dh(h, t, Vs)

dn(n, t, Vs)

dq(q, t, Ca)

ds(s, t, Vd)

get_batch_shape([batch_size])

get_delay_data(identifier, delay_step, *indices)

Get delay data according to the provided delay steps.

inf_c(Vd)

inf_h(Vs)

inf_m(Vs)

inf_n(Vs)

inf_q(Ca)

inf_s(Vd)

load_states(filename[, verbose])

Load the model states.

nodes([method, level, include_self])

Collect all children nodes.

offline_fit(target, fit_record)

offline_init()

online_fit(target, fit_record)

online_init()

register_delay(identifier, delay_step, ...)

Register delay variable.

register_implicit_nodes(*nodes, **named_nodes)

register_implicit_vars(*variables, ...)

reset([batch_size])

Reset function which reset the whole variables in the model.

reset_local_delays([nodes])

Reset local delay variables.

reset_state([batch_size])

Reset function which reset the states in the model.

save_states(filename[, variables])

Save the model states.

train_vars([method, level, include_self])

The shortcut for retrieving all trainable variables.

unique_name([name, type_])

Get the unique name for this object.

update(tdi[, x])

The function to specify the updating rule.

update_local_delays([nodes])

Update local delay variables.

vars([method, level, include_self])

Collect all variables in this node and the children nodes.

Attributes

derivative

global_delay_data

mode

Mode of the model, which is useful to control the multiple behaviors of the model.

name

Name of the model.

varshape

The shape of variables in the neuron group.