Explaining pathological changes in axonal excitability by dynamical analysis (Coggan et al. 2011)

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"... To help decipher the biophysical basis for ‘paroxysmal’ spiking, we replicated afterdischarge (i.e. continued spiking after a brief stimulus) in a minimal conductance-based axon model. ... A perturbation could abruptly switch the system between two (quasi-)stable attractor states: rest and repetitive spiking. ... Initiation of afterdischarge was explained by activation of the persistent inward current forcing the system to cross a saddle point that separates the basins of attraction associated with each attractor. Termination of afterdischarge was explained by the attractor associated with repetitive spiking being destroyed. ... The model also explains other features of paroxysmal symptoms, including temporal summation and refractoriness."
1 . Coggan JS, Ocker GK, Sejnowski TJ, Prescott SA (2011) Explaining pathological changes in axonal excitability through dynamical analysis of conductance-based models. J Neural Eng 8:065002 [PubMed]
Model Information (Click on a link to find other models with that property)
Model Type: Axon;
Brain Region(s)/Organism:
Cell Type(s):
Gap Junctions:
Simulation Environment: XPP;
Model Concept(s): Nociception;
Implementer(s): Prescott, Steven [steve.prescott at sickkids.ca]];
: modified by Jay Coggan for K diffusion

TITLE Calcium ion accumulation with longitudinal and radial diffusion

: The internal coordinate system is set up in PROCEDURE coord_cadifus()
: The scale factors set up in this procedure do not have to be recomputed
: when diam or DFree are changed.
: The amount of calcium in an annulus is ca[i]*diam^2*vol[i] with
: ca[0] being the second order correct concentration at the exact edge
: and ca[NANN-1] being the concentration at the exact center
: modified by j coggan

	SUFFIX kdifus
	:USEION k READ ko, ki, ik WRITE ki, ik
	USEION k READ ko, ki, ik WRITE ki
	RANGE ikbar
	GLOBAL vol :, Buffer0


	(molar) = (1/liter)
	(mM)	= (millimolar)
	(um)	= (micron)
	(mA)	= (milliamp)
	FARADAY = (faraday)	 (10000 coulomb)
	PI	= (pi) (1)

	DFree = .6	(um2/ms)
	diam		(um)
	ko		(mM)
	ik		(mA/cm2)
	k1buf		(/mM-ms)
	k2buf		(/ms)
	ikbar		(mA/cm2)
	ki0 = 150 	(mM)		

	ki		(mM)
	vol[NANN]	(1)	: gets extra cm2 when multiplied by diam^2

	k[NANN]	(mM) 	: k[0] is equivalent to cai
	KBuffer[NANN]  (mM)
	Buffer[NANN]    (mM)

	SOLVE state METHOD sparse
	ik = ikbar

LOCAL coord_done

	if (coord_done == 0) {
		coord_done = 1
	: note Buffer gets set to Buffer0 automatically
	: and KBuffer gets set to 0 (Default value of KBuffer0) as well
	FROM i=0 TO NANN-1 {
		k[i] = ki0

LOCAL frat[NANN] 	: gets extra cm when multiplied by diam

PROCEDURE coord() {
	LOCAL r, dr2
	: cylindrical coordinate system  with constant annuli thickness to
	: center of cell. Note however that the first annulus is half thickness
	: so that the concentration is second order correct spatially at
	: the membrane or exact edge of the cell.
	: note k[0] is at edge of cell
	:      k[NANN-1] is at center of cell
	r = 1/2
						:starts at edge (half diam)
	dr2 = r/(NANN-1)/2			:half thickness of annulus
	vol[0] = 0
	frat[0] = 2*r
	FROM i=0 TO NANN-2 {
		vol[i] = vol[i] + PI*(r-dr2/2)*2*dr2	:interior half
		r = r - dr2
		frat[i+1] = 2*PI*r/(2*dr2)	:exterior edge of annulus
					: divided by distance between centers
		r = r - dr2
		vol[i+1] = PI*(r+dr2/2)*2*dr2	:outer half of annulus

LOCAL dsq, dsqvol : can't define local variable in KINETIC block or use
KINETIC state {
	COMPARTMENT i, diam*diam*vol[i] {k KBuffer Buffer}
	LONGITUDINAL_DIFFUSION j, DFree*diam*diam*vol[j] {k}
	~ k[0] << (-ik*PI*diam*frat[0]/(FARADAY))
	FROM i=0 TO NANN-2 {
		~ k[i] <-> k[i+1] (DFree*frat[i+1], DFree*frat[i+1])
	dsq = diam*diam
	FROM i=0 TO NANN-1 {
		dsqvol = dsq*vol[i]
		~ k[i] + Buffer[i] <-> KBuffer[i] (k1buf*dsqvol,k2buf*dsqvol)
	:ki = k[3]
	ki = (k[0] + k[1] + k[2] + k[3])/4

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