BioNMDA

class brainpy.state.BioNMDA(in_size, name=None, alpha1=Quantity(2., "kHz"), beta1=Quantity(0.01, "kHz"), alpha2=Quantity(1., "1 / (mM * ms)"), beta2=Quantity(0.5, "kHz"), T=Quantity(1., "mM"), T_dur=Quantity(0.5, "ms"), g_initializer=Constant(value=0. mS), x_initializer=Constant(value=0.0))

Biological NMDA receptor synapse model.

This class implements a detailed kinetic model of NMDA (N-methyl-D-aspartate) receptor-mediated synaptic transmission. NMDA receptors are ionotropic glutamate receptors that play a crucial role in synaptic plasticity, learning, and memory.

Unlike AMPA receptors, NMDA receptors exhibit both ligand-gating and voltage-dependent properties. The voltage dependence arises from the blocking of the receptor pore by extracellular magnesium ions (Mg²⁺) at resting potential. The model uses a second-order kinetic scheme to capture the dynamics of NMDA receptors:

\[ \frac{dg}{dt} = \alpha_1 x (1-g) - \beta_1 g \]

\[ \frac{dx}{dt} = \alpha_2 [T] (1-x) - \beta_2 x \]

\[ I_{syn} = g_{max} \cdot g \cdot g_{\infty}(V,[Mg^{2+}]_o) \cdot (V - E) \]

where:

• \(g\) represents the fraction of receptors in the open state

• \(x\) is an auxiliary variable representing an intermediate state

• \(\alpha_1, \beta_1\) are the conversion rates for the \(g\) variable [ms^-1]

• \(\alpha_2, \beta_2\) are the conversion rates for the \(x\) variable [ms^-1] or [ms^-1 mM^-1]

• \([T]\) is the neurotransmitter (glutamate) concentration [mM]

• \(g_{\infty}(V,[Mg^{2+}]_o)\) is the voltage-dependent magnesium block function

• \(I_{syn}\) is the resulting synaptic current

• \(g_{max}\) is the maximum conductance

• \(V\) is the membrane potential

• \(E\) is the reversal potential

The magnesium block is typically modeled as:

\[ g_{\infty}(V,[Mg^{2+}]_o) = \frac{1}{1 + [Mg^{2+}]_o \cdot \exp(-a \cdot V) / b} \]

The neurotransmitter concentration \([T]\) follows a square pulse of amplitude T and duration T_dur after each presynaptic spike.

Parameters:

• in_size (Size) – Size of the input.

• name (str, optional) – Name of the synapse instance.

• alpha1 (ArrayLike, default 2.0/u.ms) – Conversion rate of g from inactive to active [ms^-1].

• beta1 (ArrayLike, default 0.01/u.ms) – Conversion rate of g from active to inactive [ms^-1].

• alpha2 (ArrayLike, default 1.0/(u.ms*u.mM)) – Conversion rate of x from inactive to active [ms^-1 mM^-1].

• beta2 (ArrayLike, default 0.5/u.ms) – Conversion rate of x from active to inactive [ms^-1].

• T (ArrayLike, default 1.0*u.mM) – Peak neurotransmitter concentration when released [mM].

• T_dur (ArrayLike, default 0.5*u.ms) – Duration of neurotransmitter presence in the synaptic cleft [ms].

• g_initializer (ArrayLike or Callable, default init.Constant(0. * u.mS)) – Initial value or initializer for the synaptic conductance.

• x_initializer (ArrayLike or Callable, default init.Constant(0.)) – Initial value or initializer for the auxiliary state variable.

g

Fraction of receptors in the open state.

Type:

HiddenState

x

Auxiliary state variable representing intermediate receptor state.

Type:

HiddenState

spike_arrival_time

Time of the most recent presynaptic spike.

Type:

ShortTermState

See also

AMPA

Fast excitatory AMPA receptor synapse model.

GABAa

Inhibitory GABAa receptor synapse model.

MgBlock

Voltage-dependent Mg2+ block output module.

Notes

• NMDA receptors have slower kinetics compared to AMPA receptors [1], with rise times of several milliseconds and decay time constants of tens to hundreds of milliseconds.

• The voltage-dependent magnesium block [4] is typically implemented in the output layer or postsynaptic neuron model, not in this synapse model itself.

• NMDA receptors are permeable to calcium ions, which can trigger various intracellular signaling cascades important for synaptic plasticity [3].

• This implementation uses an exponential Euler integration method for both state variables.

• For structural information about NMDA receptors, see [2].

References

[1]

Devaney, A. J. (2010). Mathematical Foundations of Neuroscience. Springer New York, 162.

[2]

Furukawa, H., Singh, S. K., Mancusso, R., & Gouaux, E. (2005). Subunit arrangement and function in NMDA receptors. Nature, 438(7065), 185-192.

[3]

Li, F., & Tsien, J. Z. (2009). Memory and the NMDA receptors. The New England Journal of Medicine, 361(3), 302.

[4]

Jahr, C. E., & Stevens, C. F. (1990). Voltage dependence of NMDA-activated macroscopic conductances predicted by single-channel kinetics. Journal of Neuroscience, 10(9), 3178-3182.

Examples

>>> import brainpy
>>> import brainstate
>>> import saiunit as u
>>> # Create an NMDA synapse
>>> nmda = brainpy.state.BioNMDA(100)
>>> nmda.init_state(batch_size=1)

init_state(batch_size=None, **kwargs)

State initialization function.

reset_state(batch_size=None, **kwargs)

State resetting function.