glong

README

Glong: longer time-scale conductances from GABA-B / GIRK and NMDA receptors

This package implements two complementary conductances, GABA-B / GIRK and NMDA, that have relatively long time constants (on the order of 100-200 msec) and thus drive these longer-scale dynamics in neurons, beyond the basic AMPA and GABA-A channels with short ~10 msec time constants. In addition to their longer time constants, these neurons have interesting mirror-image rectification properties (voltage dependencies), inward and outward, which are a critical part of their function (Sanders et al, 2013).

GABA-B / GIRK

GABA-B is an inhibitory channel activated by the usual GABA inhibitory neurotransmitter, which is coupled to the GIRK G-protein coupled inwardly rectifying potassium (K) channel. It is ubiquitous in the brain, and is likely essential for basic neural function (especially in spiking networks from a computational perspective). The inward rectification is caused by a Mg+ ion block from the inside of the neuron, which means that these channels are most open when the neuron is hyperpolarized (inactive), and thus it serves to keep inactive neurons inactive.

In standard Leabra rate-code neurons using FFFB inhibition, the continuous nature of the GABA-A type inhibition serves this function already, so these GABA-B channels have not been as important, but whenever a discrete spiking function has been used along with FFFB inhibition or direct interneuron inhibition, there is a strong tendency for every neuron to fire at some point, in a rolling fashion, because neurons that are initially inhibited during the first round of firing can just pop back up once that initial wave of associated GABA-A inhibition passes. This is especially problematic for untrained networks where excitatory connections are not well differentiated, and neurons are receiving very similar levels of excitatory input. In this case, learning does not have the ability to further differentiate the neurons, and does not work effectively.

NMDA

NMDA is predominantly important for learning in most parts of the brain, but in frontal areas especially, a particular type of NMDA receptor drives sustained active maintenance, by driving sustained excitatory voltage-gated conductances (Goldman-Rakic, 1995; Lisman et al, 1998; Brunel & Wang, 2001). The well-known external Mg+ ion block of NMDA channels, which is essential for its Hebbian character in learning, also has the beneficial effect of keeping active neurons active by virtue of causing an outward rectification that is the mirror-image of the GIRK inward rectification, with the complementary functional effect. The Brunel & Wang (2001) model is widely used in many other active maintenance / working memory models.

Sanders et al, 2013 emphasize the "perfect couple" nature of the complementary bistability of GABA-B and NMDA, and show that it produces a more robust active maintenance dynamic in frontal cortical circuits.

We are using both NMDA and GABA-B to support robust active maintenance but also the ability to clear out previously-maintained representations, as a result of the GABA-B sustained inhibition.

Implementation

Spiking effects on Vm: Backpropagating Action Potentials

Spikes in the soma of a neuron also drive backpropagating action potentials back into the dendrites, which increases the overall membrane potential there above what is produced by the raw synaptic inputs. In our rate code models, we simulate these effects by adding a proportion of the rate code activation value to the Vm value. Also note that this Vm value is an equilibrium Vm that is not reset back to rest after each spike, so it is itself perhaps a bit elevated as a result.

GABA-B

The GABA-B / GIRK dynamics are based on a combination of Sanders et al, 2013 and Thomson & Destexhe, 1999 (which was major source for Sanders et al, 2013).

T&D found a sigmoidal logistic function described the maximum conductance as a function of GABA spikes, which saturates after about 10 spikes, and a conductance time course that is fit by a 4th-order function similar to the Hodgkin Huxley equations, with a peak at about 60 msec and a total duration envelope of around 200 msec. Sanders et al implemented these dynamics directly and describe the result as such:

These equations give a latency of GABAB-activated KIR current of 􏰅10 ms, time to peak of 􏰅50 ms, and a decay time constant of 􏰅80 ms as described by Thomson and Destexhe (1999). A slow time constant for GABA removal was chosen because much of the GABAB response is mediated by extrasynaptic receptors (Kohl and Paulsen, 2010) and because the time constant for GABA decline at extracellular sites appears to be much slower than at synaptic sites.

We take a simpler approach which is to just use a bi-exponential function to simulate the time course, using 45 and 50 msec time constants.

See gabab_plot subdirectory for program that generates these plots:

V-G plot for GABA-B / GIRK from Sanders et al, 2013.

Spike-G sigmoid plot for GABA-B / GIRK from Thomson & Destexhe (1999)

Bi-exponential time course with 45msec rise and 50msec decay fitting to results from Thomson & Destexhe (1999)

NMDA

The NMDA code uses the exponential function from Brunel & Wang (2001), which is very similar in shape to the Sanders et al, 2013 curve, to compute the voltage dependency of the current:

	vbio := v*100 - 100
	return 1 / (1 + 0.28*math32.Exp(-0.062*vbio))

where v is the normalized Vm value from the Leabra rate code equations -- vbio converts it into biological units.

See the nmda_plot directory for a program that plots this function, which looks like this (in terms of net conductance also taking into account the driving potential, which adds an (E - v) term in the numerator).

V-G plot for NMDA from Brunel & Wang (2001)

Documentation

Index

• type GABABParams
  • func (gp *GABABParams) BiExp(g, x float32) (dG, dX float32)
  • func (gp *GABABParams) Defaults()
  • func (gp *GABABParams) GABAB(gabaB, gabaBx, gi float32) (g, x float32)
  • func (gp *GABABParams) GFmS(s float32) float32
  • func (gp *GABABParams) GFmV(v float32) float32
  • func (gp *GABABParams) GgabaB(gabaB, vm float32) float32
  • func (gp *GABABParams) Update()
• type NMDAParams
  • func (np *NMDAParams) Defaults()
  • func (np *NMDAParams) GFmV(v float32) float32
  • func (np *NMDAParams) Gnmda(nmda, vm float32) float32
  • func (np *NMDAParams) NMDA(nmda, geraw, nmdaSyn float32) float32
• type VGCCParams
  • func (np *VGCCParams) DMHFmV(v, m, h float32) (float32, float32)
  • func (np *VGCCParams) Defaults()
  • func (np *VGCCParams) GFmV(v float32) float32
  • func (np *VGCCParams) Gvgcc(vm, m, h float32) float32
  • func (np *VGCCParams) HFmV(vbio float32) float32
  • func (np *VGCCParams) MFmV(vbio float32) float32

Types

type GABABParams

type GABABParams struct {
	RiseTau  float32 `def:"45" desc:"rise time for bi-exponential time dynamics of GABA-B"`
	DecayTau float32 `def:"50" desc:"decay time for bi-exponential time dynamics of GABA-B"`
	Gbar     float32 `def:"0.2" desc:"overall strength multiplier of GABA-B current"`
	Gbase    float32 `` /* 130-byte string literal not displayed */
	GiSpike  float32 `def:"10" desc:"multiplier for converting Gi to equivalent GABA spikes"`
	MaxTime  float32 `inactive:"+" desc:"time offset when peak conductance occurs, in msec, computed from RiseTau and DecayTau"`
	TauFact  float32 `view:"-" desc:"time constant factor used in integration: (Decay / Rise) ^ (Rise / (Decay - Rise))"`
}

GABABParams control the GABAB dynamics in PFC Maint neurons, based on Brunel & Wang (2001) parameters.

func (*GABABParams) BiExp

func (gp *GABABParams) BiExp(g, x float32) (dG, dX float32)

BiExp computes bi-exponential update, returns dG and dX deltas to add to g and x

func (*GABABParams) Defaults

func (gp *GABABParams) Defaults()

func (*GABABParams) GABAB

func (gp *GABABParams) GABAB(gabaB, gabaBx, gi float32) (g, x float32)

GABAB returns the updated GABA-B / GIRK activation and underlying x value based on current values and gi inhibitory conductance (proxy for GABA spikes)

func (*GABABParams) GFmS

func (gp *GABABParams) GFmS(s float32) float32

GFmS returns the GABA-B conductance as a function of GABA spiking rate, based on normalized spiking factor (i.e., Gi from FFFB etc)

func (*GABABParams) GFmV

func (gp *GABABParams) GFmV(v float32) float32

GFmV returns the GABA-B conductance as a function of normalized membrane potential

func (*GABABParams) GgabaB

func (gp *GABABParams) GgabaB(gabaB, vm float32) float32

GgabaB returns the overall net GABAB / GIRK conductance including Gbar, Gbase, and voltage-gating

func (*GABABParams) Update

func (gp *GABABParams) Update()

type NMDAParams

type NMDAParams struct {
	GeTot float32 `` /* 185-byte string literal not displayed */
	Tau   float32 `def:"100" desc:"decay time constant for NMDA current -- rise time is 2 msec and not worth extra effort for biexponential"`
	Gbar  float32 `` /* 131-byte string literal not displayed */
}

NMDAParams control the NMDA dynamics, based on Brunel & Wang (2001) parameters.

func (*NMDAParams) Defaults

func (np *NMDAParams) Defaults()

func (*NMDAParams) GFmV

func (np *NMDAParams) GFmV(v float32) float32

GFmV returns the NMDA conductance as a function of normalized membrane potential

func (*NMDAParams) Gnmda

func (np *NMDAParams) Gnmda(nmda, vm float32) float32

Gnmda returns the NMDA net conductance from nmda activation and vm

func (*NMDAParams) NMDA

func (np *NMDAParams) NMDA(nmda, geraw, nmdaSyn float32) float32

NMDA returns the updated NMDA activation from current NMDA, GeRaw, and NMDASyn input

type VGCCParams

type VGCCParams struct {
	Gbar float32 `def:"0.01" desc:"strength of VGCC current"`
}

VGCCParams control the standard L-type Ca channel

func (*VGCCParams) DMHFmV

func (np *VGCCParams) DMHFmV(v, m, h float32) (float32, float32)

DMHFmV returns the change in M, H factors as a function of V normalized (0-1)

func (*VGCCParams) Defaults

func (np *VGCCParams) Defaults()

func (*VGCCParams) GFmV

func (np *VGCCParams) GFmV(v float32) float32

GFmV returns the VGCC conductance as a function of normalized membrane potential

func (*VGCCParams) Gvgcc

func (np *VGCCParams) Gvgcc(vm, m, h float32) float32

Gvgcc returns the VGCC net conductance from m, h activation and vm

func (*VGCCParams) HFmV

func (np *VGCCParams) HFmV(vbio float32) float32

HFmV returns the H gate function from vbio (not normalized, must not exceed 0)

func (*VGCCParams) MFmV

func (np *VGCCParams) MFmV(vbio float32) float32

MFmV returns the M gate function from vbio (not normalized, must not exceed 0)