# brainmodels.synapses.NMDA

class brainmodels.synapses.NMDA(pre, post, conn, delay=0.0, g_max=0.15, E=0.0, cc_Mg=1.2, alpha=0.062, beta=3.57, tau_decay=100.0, a=0.5, tau_rise=2.0, method='exponential_euler', **kwargs)[source]

Conductance-based NMDA synapse model.

Model Descriptions

The NMDA receptor is a glutamate receptor and ion channel found in neurons. The NMDA receptor is one of three types of ionotropic glutamate receptors, the other two being AMPA and kainate receptors.

The NMDA receptor mediated conductance depends on the postsynaptic voltage. The voltage dependence is due to the blocking of the pore of the NMDA receptor from the outside by a positively charged magnesium ion. The channel is nearly completely blocked at resting potential, but the magnesium block is relieved if the cell is depolarized. The fraction of channels $$g_{\infty}$$ that are not blocked by magnesium can be fitted to

$g_{\infty}(V,[{Mg}^{2+}]_{o}) = (1+{e}^{-\alpha V} \frac{[{Mg}^{2+}]_{o}} {\beta})^{-1}$

Here $$[{Mg}^{2+}]_{o}$$ is the extracellular magnesium concentration, usually 1 mM. Thus, the channel acts as a “coincidence detector” and only once both of these conditions are met, the channel opens and it allows positively charged ions (cations) to flow through the cell membrane 2.

If we make the approximation that the magnesium block changes instantaneously with voltage and is independent of the gating of the channel, the net NMDA receptor-mediated synaptic current is given by

$I_{syn} = g_{NMDA}(t) (V(t)-E) \cdot g_{\infty}$

where $$V(t)$$ is the post-synaptic neuron potential, $$E$$ is the reversal potential.

Simultaneously, the kinetics of synaptic state $$g$$ is given by

$\begin{split}& g_{NMDA} (t) = g_{max} g \\ & \frac{d g}{dt} = -\frac{g} {\tau_{decay}}+a x(1-g) \\ & \frac{d x}{dt} = -\frac{x}{\tau_{rise}}+ \sum_{k} \delta(t-t_{j}^{k})\end{split}$

where the decay time of NMDA currents is usually taken to be $$\tau_{decay}$$ =100 ms, $$a= 0.5 ms^{-1}$$, and $$\tau_{rise}$$ =2 ms.

The NMDA receptor has been thought to be very important for controlling synaptic plasticity and mediating learning and memory functions 3.

Model Examples

Model Parameters

 Parameter Init Value Unit Explanation delay 0 ms The decay length of the pre-synaptic spikes. g_max .15 µmho(µS) The synaptic maximum conductance. E 0 mV The reversal potential for the synaptic current. alpha .062 Binding constant. beta 3.57 Unbinding constant. cc_Mg 1.2 mM Concentration of Magnesium ion. tau_decay 100 ms The time constant of the synaptic decay phase. tau_rise 2 ms The time constant of the synaptic rise phase. a .5 1/ms

Model Variables

 Member name Initial values Explanation g 0 Synaptic conductance. x 0 Synaptic gating variable. pre_spike False The history spiking states of the pre-synaptic neurons.

References

1

Brunel N, Wang X J. Effects of neuromodulation in a cortical network model of object working memory dominated by recurrent inhibition[J]. Journal of computational neuroscience, 2001, 11(1): 63-85.

2

Furukawa, Hiroyasu, Satinder K. Singh, Romina Mancusso, and Eric Gouaux. “Subunit arrangement and function in NMDA receptors.” Nature 438, no. 7065 (2005): 185-192.

3

Li, F. and Tsien, J.Z., 2009. Memory and the NMDA receptors. The New England journal of medicine, 361(3), p.302.

4

https://en.wikipedia.org/wiki/NMDA_receptor

__init__(pre, post, conn, delay=0.0, g_max=0.15, E=0.0, cc_Mg=1.2, alpha=0.062, beta=3.57, tau_decay=100.0, a=0.5, tau_rise=2.0, method='exponential_euler', **kwargs)[source]

Methods

 __init__(pre, post, conn[, delay, g_max, E, ...]) build_inputs([inputs, show_code]) build_monitors([show_code]) cpu() cuda() derivative(g, x, t) ints([method]) Collect all integrators in this node and the children nodes. jax_update(_t, _dt) load_states(filename[, verbose, check]) Load the model states. nodes([method, _paths]) Collect all children nodes. numpy_update(_t, _dt) register_constant_delay(key, size, delay[, ...]) Register a constant delay. run(duration[, dt, report, inputs, extra_func]) The running function. save_states(filename[, all_vars]) Save the model states. step(t_and_dt, **kwargs) to(devices) tpu() train_vars([method]) The shortcut for retrieving all trainable variables. unique_name([name, type]) Get the unique name for this object. update(*args, **kwargs) The function to specify the updating rule. vars([method]) Collect all variables in this node and the children nodes.

Attributes

 implicit_nodes Used to wrap the implicit children nodes which cannot be accessed by self.xxx implicit_vars Used to wrap the implicit variables which cannot be accessed by self.xxx target_backend Used to specify the target backend which the model to run.