brainmodels.synapses module

AMPA(pre, post, conn[, delay, g_max, E, ...])

AMPA conductance-based synapse model.

AlphaCOBA(pre, post, conn[, delay, g_max, ...])

Conductance-based alpha synapse model.

AlphaCUBA(pre, post, conn[, delay, g_max, ...])

Current-based alpha synapse model.

DualExpCOBA(pre, post, conn[, delay, g_max, ...])

Conductance-based dual exponential synapse model.

DualExpCUBA(pre, post, conn[, delay, g_max, ...])

Current-based dual exponential synapse model.

ExpCOBA(pre, post, conn[, g_max, delay, ...])

Conductance-based exponential decay synapse model.

ExpCUBA(pre, post, conn[, g_max, delay, ...])

Current-based exponential decay synapse model.

GABAa(pre, post, conn[, delay, g_max, E, ...])

GABAa conductance-based synapse model.

NMDA(pre, post, conn[, delay, g_max, E, ...])

Conductance-based NMDA synapse model.

STP(pre, post, conn[, U, tau_f, tau_d, tau, ...])

Short-term plasticity model.

Synapse(pre, post, conn[, method])

VoltageJump(pre, post, conn[, delay, ...])

Voltage jump synapse model.

class brainmodels.synapses.AMPA(pre, post, conn, delay=0.0, g_max=0.42, E=0.0, alpha=0.98, beta=0.18, T=0.5, T_duration=0.5, method='exponential_euler', **kwargs)[source]

AMPA conductance-based synapse model.

Model Descriptions

AMPA receptor is an ionotropic receptor, which is an ion channel. When it is bound by neurotransmitters, it will immediately open the ion channel, causing the change of membrane potential of postsynaptic neurons.

A classical model is to use the Markov process to model ion channel switch. Here \(g\) represents the probability of channel opening, \(1-g\) represents the probability of ion channel closing, and \(\alpha\) and \(\beta\) are the transition probability. Because neurotransmitters can open ion channels, the transfer probability from \(1-g\) to \(g\) is affected by the concentration of neurotransmitters. We denote the concentration of neurotransmitters as \([T]\) and get the following Markov process.

../images/synapse_markov.png

We obtained the following formula when describing the process by a differential equation.

\[\frac{ds}{dt} =\alpha[T](1-g)-\beta g\]

where \(\alpha [T]\) denotes the transition probability from state \((1-g)\) to state \((g)\); and \(\beta\) represents the transition probability of the other direction. \(\alpha\) is the binding constant. \(\beta\) is the unbinding constant. \([T]\) is the neurotransmitter concentration, and has the duration of 0.5 ms.

Moreover, the post-synaptic current on the post-synaptic neuron is formulated as

\[I_{syn} = g_{max} g (V-E)\]

where \(g_{max}\) is the maximum conductance, and E is the reverse potential.

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

g_max

.42

µmho(µS)

Maximum conductance.

E

0

mV

The reversal potential for the synaptic current.

alpha

.98

Binding constant.

beta

.18

Unbinding constant.

T

.5

mM

Neurotransmitter concentration.

T_duration

.5

ms

Duration of the neurotransmitter concentration.

Model Variables

Member name

Initial values

Explanation

g

0

Synapse gating variable.

pre_spike

False

The history of pre-synaptic neuron spikes.

spike_arrival_time

-1e7

The arrival time of the pre-synaptic neuron spike.

References

1

Vijayan S, Kopell N J. Thalamic model of awake alpha oscillations and implications for stimulus processing[J]. Proceedings of the National Academy of Sciences, 2012, 109(45): 18553-18558.

class brainmodels.synapses.AlphaCOBA(pre, post, conn, delay=0.0, g_max=1.0, tau_decay=10.0, E=0.0, method='exponential_euler', **kwargs)[source]

Conductance-based alpha synapse model.

Model Descriptions

The conductance-based alpha synapse model is similar with the current-based alpha synapse model, except the expression which output onto the post-synaptic neurons:

\[I_{syn}(t) = g_{\mathrm{syn}}(t) (V(t)-E)\]

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

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

tau_decay

2

ms

The decay time constant of the synaptic state.

g_max

.2

µmho(µS)

The maximum conductance.

E

0

mV

The reversal potential for the synaptic current.

Model Variables

Variables name

Initial Value

Explanation

g

0

Synapse conductance on the post-synaptic neuron.

h

0

Gating variable.

pre_spike

False

The history spiking states of the pre-synaptic neurons.

References

1

Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw. “The Synapse.” Principles of Computational Modelling in Neuroscience. Cambridge: Cambridge UP, 2011. 172-95. Print.

class brainmodels.synapses.AlphaCUBA(pre, post, conn, delay=0.0, g_max=1.0, tau_decay=10.0, method='exponential_euler', **kwargs)[source]

Current-based alpha synapse model.

Model Descriptions

The analytical expression of alpha synapse is given by:

\[g_{syn}(t)= g_{max} \frac{t-t_{s}}{\tau} \exp \left(-\frac{t-t_{s}}{\tau}\right).\]

While, this equation is hard to implement. So, let’s try to convert it into the differential forms:

\[\begin{split}\begin{aligned} &g_{\mathrm{syn}}(t)= g_{\mathrm{max}} g \\ &\frac{d g}{d t}=-\frac{g}{\tau}+h \\ &\frac{d h}{d t}=-\frac{h}{\tau}+\delta\left(t_{0}-t\right) \end{aligned}\end{split}\]

The current onto the post-synaptic neuron is given by

\[I_{syn}(t) = g_{\mathrm{syn}}(t).\]

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

tau_decay

2

ms

The decay time constant of the synaptic state.

g_max

.2

µmho(µS)

The maximum conductance.

Model Variables

Variables name

Initial Value

Explanation

g

0

Synapse conductance on the post-synaptic neuron.

h

0

Gating variable.

pre_spike

False

The history spiking states of the pre-synaptic neurons.

References

1

Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw. “The Synapse.” Principles of Computational Modelling in Neuroscience. Cambridge: Cambridge UP, 2011. 172-95. Print.

class brainmodels.synapses.DualExpCOBA(pre, post, conn, delay=0.0, g_max=1.0, tau_decay=10.0, tau_rise=1.0, E=0.0, method='exponential_euler', **kwargs)[source]

Conductance-based dual exponential synapse model.

Model Descriptions

The conductance-based dual exponential synapse model is similar with the current-based dual exponential synapse model, except the expression which output onto the post-synaptic neurons:

\[I_{syn}(t) = g_{\mathrm{syn}}(t) (V(t)-E)\]

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

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

tau_decay

10

ms

The time constant of the synaptic decay phase.

tau_rise

1

ms

The time constant of the synaptic rise phase.

g_max

1

µmho(µS)

The maximum conductance.

E

0

mV

The reversal potential for the synaptic current.

Model Variables

Member name

Initial values

Explanation

g

0

Synapse conductance on the post-synaptic neuron.

s

0

Gating variable.

pre_spike

False

The history spiking states of the pre-synaptic neurons.

References

1

Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw. “The Synapse.” Principles of Computational Modelling in Neuroscience. Cambridge: Cambridge UP, 2011. 172-95. Print.

class brainmodels.synapses.DualExpCUBA(pre, post, conn, delay=0.0, g_max=1.0, tau_decay=10.0, tau_rise=1.0, method='exponential_euler', **kwargs)[source]

Current-based dual exponential synapse model.

Model Descriptions

The dual exponential synapse model [1]_, also named as difference of two exponentials model, is given by:

\[g_{\mathrm{syn}}(t)=g_{\mathrm{max}} \frac{\tau_{1} \tau_{2}}{ \tau_{1}-\tau_{2}}\left(\exp \left(-\frac{t-t_{0}}{\tau_{1}}\right) -\exp \left(-\frac{t-t_{0}}{\tau_{2}}\right)\right)\]

where \(\tau_1\) is the time constant of the decay phase, \(\tau_2\) is the time constant of the rise phase, \(t_0\) is the time of the pre-synaptic spike, \(g_{\mathrm{max}}\) is the maximal conductance.

However, in practice, this formula is hard to implement. The equivalent solution is two coupled linear differential equations [2]_:

\[\begin{split}\begin{aligned} &g_{\mathrm{syn}}(t)=g_{\mathrm{max}} g \\ &\frac{d g}{d t}=-\frac{g}{\tau_{\mathrm{decay}}}+h \\ &\frac{d h}{d t}=-\frac{h}{\tau_{\text {rise }}}+ \delta\left(t_{0}-t\right), \end{aligned}\end{split}\]

The current onto the post-synaptic neuron is given by

\[I_{syn}(t) = g_{\mathrm{syn}}(t).\]

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

tau_decay

10

ms

The time constant of the synaptic decay phase.

tau_rise

1

ms

The time constant of the synaptic rise phase.

g_max

1

µmho(µS)

The maximum conductance.

Model Variables

Member name

Initial values

Explanation

g

0

Synapse conductance on the post-synaptic neuron.

s

0

Gating variable.

pre_spike

False

The history spiking states of the pre-synaptic neurons.

References

1

Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw. “The Synapse.” Principles of Computational Modelling in Neuroscience. Cambridge: Cambridge UP, 2011. 172-95. Print.

2

Roth, A., & Van Rossum, M. C. W. (2009). Modeling Synapses. Computational Modeling Methods for Neuroscientists.

class brainmodels.synapses.ExpCOBA(pre, post, conn, g_max=1.0, delay=0.0, tau=8.0, E=0.0, method='exponential_euler', **kwargs)[source]

Conductance-based exponential decay synapse model.

Model Descriptions

The conductance-based exponential decay synapse model is similar with the current-based exponential decay synapse model, except the expression which output onto the post-synaptic neurons:

\[I_{syn}(t) = g_{\mathrm{syn}}(t) (V(t)-E)\]

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

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

tau_decay

8

ms

The time constant of decay.

g_max

1

µmho(µS)

The maximum conductance.

E

0

mV

The reversal potential for the synaptic current.

Model Variables

Member name

Initial values

Explanation

g

0

Gating variable.

pre_spike

False

The history spiking states of the pre-synaptic neurons.

References

1

Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw. “The Synapse.” Principles of Computational Modelling in Neuroscience. Cambridge: Cambridge UP, 2011. 172-95. Print.

class brainmodels.synapses.ExpCUBA(pre, post, conn, g_max=1.0, delay=0.0, tau=8.0, method='exponential_euler', **kwargs)[source]

Current-based exponential decay synapse model.

Model Descriptions

The single exponential decay synapse model assumes the release of neurotransmitter, its diffusion across the cleft, the receptor binding, and channel opening all happen very quickly, so that the channels instantaneously jump from the closed to the open state. Therefore, its expression is given by

\[g_{\mathrm{syn}}(t)=g_{\mathrm{max}} e^{-\left(t-t_{0}\right) / \tau}\]

where \(\tau_{delay}\) is the time constant of the synaptic state decay, \(t_0\) is the time of the pre-synaptic spike, \(g_{\mathrm{max}}\) is the maximal conductance.

Accordingly, the differential form of the exponential synapse is given by

\[\begin{split}\begin{aligned} & g_{\mathrm{syn}}(t) = g_{max} g \\ & \frac{d g}{d t} = -\frac{g}{\tau_{decay}}+\sum_{k} \delta(t-t_{j}^{k}). \end{aligned}\end{split}\]

For the current output onto the post-synaptic neuron, its expression is given by

\[I_{\mathrm{syn}}(t) = g_{\mathrm{syn}}(t)\]

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

tau_decay

8

ms

The time constant of decay.

g_max

1

µmho(µS)

The maximum conductance.

Model Variables

Member name

Initial values

Explanation

g

0

Gating variable.

pre_spike

False

The history spiking states of the pre-synaptic neurons.

References

1

Sterratt, David, Bruce Graham, Andrew Gillies, and David Willshaw. “The Synapse.” Principles of Computational Modelling in Neuroscience. Cambridge: Cambridge UP, 2011. 172-95. Print.

class brainmodels.synapses.GABAa(pre, post, conn, delay=0.0, g_max=0.04, E=- 80.0, alpha=0.53, beta=0.18, T=1.0, T_duration=1.0, method='exponential_euler', **kwargs)[source]

GABAa conductance-based synapse model.

Model Descriptions

GABAa synapse model has the same equation with the AMPA synapse,

\[\begin{split}\frac{d g}{d t}&=\alpha[T](1-g) - \beta g \\ I_{syn}&= - g_{max} g (V - E)\end{split}\]

but with the difference of:

  • Reversal potential of synapse \(E\) is usually low, typically -80. mV

  • Activating rate constant \(\alpha=0.53\)

  • De-activating rate constant \(\beta=0.18\)

  • Transmitter concentration \([T]=1\,\mu ho(\mu S)\) when synapse is triggered by a pre-synaptic spike, with the duration of 1. ms.

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

delay

0

ms

The decay length of the pre-synaptic spikes.

g_max

0.04

µmho(µS)

Maximum synapse conductance.

E

-80

mV

Reversal potential of synapse.

alpha

0.53

Activating rate constant of G protein catalyzed by activated GABAb receptor.

beta

0.18

De-activating rate constant of G protein.

T

1

mM

Transmitter concentration when synapse is triggered by a pre-synaptic spike.

T_duration

1

ms

Transmitter concentration duration time after being triggered.

Model Variables

Member name

Initial values

Explanation

g

0

Synapse gating variable.

pre_spike

False

The history of pre-synaptic neuron spikes.

spike_arrival_time

-1e7

The arrival time of the pre-synaptic neuron spike.

References

1

Destexhe, Alain, and Denis Paré. “Impact of network activity on the integrative properties of neocortical pyramidal neurons in vivo.” Journal of neurophysiology 81.4 (1999): 1531-1547.

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

class brainmodels.synapses.STP(pre, post, conn, U=0.15, tau_f=1500.0, tau_d=200.0, tau=8.0, A=1.0, delay=0.0, method='exponential_euler', **kwargs)[source]

Short-term plasticity model.

Model Descriptions

Short-term plasticity (STP) [1]_ [2]_ [3]_, also called dynamical synapses, refers to a phenomenon in which synaptic efficacy changes over time in a way that reflects the history of presynaptic activity. Two types of STP, with opposite effects on synaptic efficacy, have been observed in experiments. They are known as Short-Term Depression (STD) and Short-Term Facilitation (STF).

In the model proposed by Tsodyks and Markram [4]_ 5, the STD effect is modeled by a normalized variable \(x (0 \le x \le 1)\), denoting the fraction of resources that remain available after neurotransmitter depletion. The STF effect is modeled by a utilization parameter \(u\), representing the fraction of available resources ready for use (release probability). Following a spike,

  • (i) \(u\) increases due to spike-induced calcium influx to the presynaptic terminal, after which

  • (ii) a fraction \(u\) of available resources is consumed to produce the post-synaptic current.

Between spikes, \(u\) decays back to zero with time constant \(\tau_f\) and \(x\) recovers to 1 with time constant \(\tau_d\).

In summary, the dynamics of STP is given by

\[\begin{split}\begin{aligned} \frac{du}{dt} & = -\frac{u}{\tau_f}+U(1-u^-)\delta(t-t_{sp}),\nonumber \\ \frac{dx}{dt} & = \frac{1-x}{\tau_d}-u^+x^-\delta(t-t_{sp}), \\ \frac{dI}{dt} & = -\frac{I}{\tau_s} + Au^+x^-\delta(t-t_{sp}), \end{aligned}\end{split}\]

where \(t_{sp}\) denotes the spike time and \(U\) is the increment of \(u\) produced by a spike. \(u^-, x^-\) are the corresponding variables just before the arrival of the spike, and \(u^+\) refers to the moment just after the spike. The synaptic current generated at the synapse by the spike arriving at \(t_{sp}\) is then given by

\[\Delta I(t_{spike}) = Au^+x^-\]

where \(A\) denotes the response amplitude that would be produced by total release of all the neurotransmitter (\(u=x=1\)), called absolute synaptic efficacy of the connections.

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

tau_d

200

ms

Time constant of short-term depression.

tau_f

1500

ms

Time constant of short-term facilitation.

U

.15

The increment of \(u\) produced by a spike.

A

1

The response amplitude that would be produced by total release of all the neurotransmitter

delay

0

ms

The decay time of the current \(I\) output onto the post-synaptic neuron groups.

Model Variables

Member name

Initial values

Explanation

u

0

Release probability of the neurotransmitters.

x

1

A Normalized variable denoting the fraction of remain neurotransmitters.

I

0

Synapse current output onto the post-synaptic neurons.

References

1

Stevens, Charles F., and Yanyan Wang. “Facilitation and depression at single central synapses.” Neuron 14, no. 4 (1995): 795-802.

2

Abbott, Larry F., J. A. Varela, Kamal Sen, and S. B. Nelson. “Synaptic depression and cortical gain control.” Science 275, no. 5297 (1997): 221-224.

3

Abbott, L. F., and Wade G. Regehr. “Synaptic computation.” Nature 431, no. 7010 (2004): 796-803.

4

Tsodyks, Misha, Klaus Pawelzik, and Henry Markram. “Neural networks with dynamic synapses.” Neural computation 10.4 (1998): 821-835.

5

Tsodyks, Misha, and Si Wu. “Short-term synaptic plasticity.” Scholarpedia 8, no. 10 (2013): 3153.

class brainmodels.synapses.Synapse(pre, post, conn, method='euler', **kwargs)[source]
class brainmodels.synapses.VoltageJump(pre, post, conn, delay=0.0, post_has_ref=False, w=1.0, post_key='V', **kwargs)[source]

Voltage jump synapse model.

Model Descriptions

\[I_{syn} (t) = \sum_{j\in C} w \delta(t-t_j-D)\]

where \(w\) denotes the chemical synaptic strength, \(t_j\) the spiking moment of the presynaptic neuron \(j\), \(C\) the set of neurons connected to the post-synaptic neuron, and \(D\) the transmission delay of chemical synapses. For simplicity, the rise and decay phases of post-synaptic currents are omitted in this model.

Model Examples

Model Parameters

Parameter

Init Value

Unit

Explanation

w

1

mV

The synaptic strength.