Anoxic depolarization, recovery: effect of brain regions and extracellular space (Hubel et al. 2016)

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Accession:187213
The extent of anoxic depolarization (AD), the initial electrophysiological event during ischemia, determines the degree of brain region-specific neuronal damage. Neurons in higher brain regions have stronger ADs and are more easily injured than neurons in lower brain region. The mechanism leading to such differences is not clear. We use a computational model based on a Hodgkin-Huxley framework which includes neural spiking dynamics, processes of ion accumulation, and homeostatic mechanisms like vascular coupling and Na/K-exchange pumps. We show that a large extracellular space (ECS) explains the recovery failure in high brain regions. A phase-space analysis shows that with a large ECS recovery from AD through potassium regulation is impossible. The code 'time_series.ode' can be used to simulate AD for a large and a small ECS and show the different behaviors. The code ‘continuations.ode’ can be used to show the fixed point structure. Depending on our choice of large or small ECS the fixed point curve implies the presence/absence of a recovery threshold that defines the potassium clearance demand.
Reference:
1 . Hübel N, Andrew RD, Ullah G (2016) Large extracellular space leads to neuronal susceptibility to ischemic injury in a Na+/K+ pumps-dependent manner. J Comput Neurosci 40:177-92 [PubMed]
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Model Information (Click on a link to find other models with that property)
Model Type: Neuron or other electrically excitable cell;
Brain Region(s)/Organism:
Cell Type(s): Abstract single compartment conductance based cell;
Channel(s): I Chloride; I Na,t; I K; I K,leak; I h; I Sodium; I Potassium; I_K,Na; Na/K pump; I Cl, leak; I Na, leak;
Gap Junctions:
Receptor(s):
Gene(s):
Transmitter(s):
Simulation Environment: XPP;
Model Concept(s): Action Potentials; Pathophysiology; Sodium pump; Depolarization block; Homeostasis; Potassium buffering;
Implementer(s):
Search NeuronDB for information about:  I Chloride; I Na,t; I K; I K,leak; I h; I Sodium; I Potassium; I_K,Na; Na/K pump; I Cl, leak; I Na, leak;
/
Huebel_Andrew_Ullah_2015
readme.txt
continuations.ode
time_series.ode
                            
###############################################################################################
# Code for fixed point continuations/bifurcation diagrams used in                             #
# "Large Extracellular Space Leads to [...] Ischemic Injury [...]" by Hubel, Ullah and Andrew #
###############################################################################################


##############################################################################################################
##############################################################################################################
###													   ###
###     The fixed point curves from Figs. 4 and 5 can be obtained as follows:				   ###
###     												   ###
###     1.) open file with XPPAUT									   ###
###     2.) run simulation twice to make sure the system is in its fixed point:				   ###
###         click "Initialconds" + "(G)o"; "Initialconds" + "(L)ast"					   ###
###         (keyboard shortcuts: "I" + "G", "I" + "L")							   ###
###     3.) open AUTO interface:									   ###
###         click "File" + "Auto"									   ###
###         (keyboard shortcut: "F" + "A")								   ###
###     4.) run 'forward' fixed point continuation:							   ###
###         click "Run" + "Steady state"								   ###
###         (keyboard shortcut: "R" + "S")								   ###
###     5.) grab point to start 'backward' continuation if desired:					   ###
###         click "Grab", then navigate along the curve with "tab" key, press "enter" to choose point 	   ###
###     6.) set continuation step size to negative:							   ###
###         click "Numerics" and change 'Ds:0.002' to 'Ds:-0.002', click "Ok"				   ###
###     7.) run backward coninuation by clicking "Run"							   ###
###     8.) save the fixed point curves and bifurcation information:					   ###
###         click "File" + "All info", choose a filename and click "Ok"					   ###
###     												   ###
###     Remark for Fig. 4:										   ###
###     Start from the 'physiological' (i.e. upper set of) initial conditions and 'par vle=720'.	   ###
###     Change 'PARMIN=-250' to 'PARMIN=-50' and accordingly 'XAUTOMIN=-250' to 'XAUTOMIN=-50'.		   ###
###     For more negative parameter values the continuation produces convergence errors "MX" at		   ###
###     extreme negative values of 'V'									   ###
###     												   ###
###     Remark for Fig. 5a:										   ###
###     Start from the 'FES' (i.e. lower set of) initial conditions and 'par vle=3700' to obntain	   ###
###     the upper branch of the continuation. Use 'physiological' initial conditions and 'par vle=3700'	   ###
###     to obtain the lower loop of the bifurcation diagram						   ###
###													   ###
##############################################################################################################
##############################################################################################################


####################################################################################
# membrane potential                                    'V'       in  [mV]	   #
# gating variables                                      'n/h'     in  [1]	   #
# intracellular ion concentrations                      'ki,cli'  in  [mM=mMol/l]  #
# rates of change                                       'X_DOT'   in  [.../msec]   #
# (factor 1000. converts to seconds)						   #
####################################################################################
V'     = 1000. * V_DOT
n'     = 1000. * N_DOT
h'     = 1000. * H_DOT

ki'    = 1000. * KI_DOT
cli'   = 1000. * CLI_DOT

######################
# Initial conditions #
######################
# physiological #
#################
# init v=-67.056664
# init n=0.070174225
# init h=0.97820824
# init ki=128.56937
# init cli=10.061391
######################
# FES for 'vle=3700' #
######################
init v=-20.288656
init n=0.66411209
init h=0.057942949
init ki=77.408791
init cli=47.911449

#########################################################################
# MAIN BIFURCATION PARAMETER:                                           #
# amount of potassium exchanged with external reservoir 'dnk' in [fmol]	#
#########################################################################
par dnk=0

########################################
# Extracellular volume 'vle' in [um^3] #
########################################
# par vle=720
par vle=3700

##################################
# Pump strength 'max_p' constant #
##################################
max_p = 6.8

###################################
# Hodgkin-Huxley gating functions #
###################################
AN  = 0.01 * (v + 34.0) / (1.0 - exp(-0.1 * (v + 34.0))) 
BN  = 0.125 * exp(-(v + 44.0) / 80.0)
AM  = 0.1 * (v + 30.0) / (1.0 - exp(-0.1 * (v + 30.0))) 
BM  = 4.0 * exp(-(v + 55.0) / 18.0) 
AH  = 0.07 * exp(-(v + 44.0) / 20)
BH  = 1.0 / (1.0 + exp(-0.1 * (v + 14.0)))
M   = AM / (AM + BM)

#####################################
# ion concentrations in [mM=mMol/l] #
#####################################################################
# intracellular sodium 'NAI' from electroneutrality                 #
# extracellular concentrations 'NAE,KE,CLE' from mass conservation  #
# normal resting values 'ki0,ke0 ...' given                         #
#####################################################################
vli    = 2160
ki0    = 128.56935
ke0    = 3.9962559
nai0   = 25.279156
nae0   = 126.84917
cli0   = 10.055541
cle0   = 124.71021

NAI = nai0 +  ki0 - ki - cli0 + cli
NAE = nae0 + (nai0 - nai) * vli/vle
KE  = ke0  + (ki0  - ki ) * vli/vle + dnk/vle * 1e3
CLE = cle0 + (cli0 - cli) * vli/vle

#############################
# Nernst potentials in [mV] #
#############################
EK  = 26.64 * log(ke  / ki)
ENA = 26.64 * log(nae / nai)
ECL =-26.64 * log(cle / cli)

############################################################################
# different types of 'l'eak and 'g'ated currents 'I(ION)_l/g' in [uA/cm^2] #
# different channel conductances                 'g(ion)_l/g' in [mS/cm^2] #
# Na/K-exchange pump current                     'IP'         in [uA/cm^2] #                             
############################################################################
gna_l  = 0.0175
gna_g  = 100.
gk_l   = 0.05
gk_g   = 40.
gcl_l  = 0.05

INA_l = gna_l            * (v - ENA)
INA_g = gna_g * M**3 * h * (v - ENA)
IK_l  = gk_l             * (v - EK)
IK_g  = gk_g * n**4      * (v - EK)
ICL_l = gcl_l            * (v - ECL)
IP    = max_p / (1.0 + exp((25 - nai)/3.)) / (1. + exp(5.5 - ke))

INA   = INA_l + INA_g + 3. * IP
IK    = IK_l  + IK_g  - 2. * IP

#############################
# Full list of change rates #
##########################################################
# membrane capacitance              'C'    in [uF/cm^2]	 #
# conversion factor                 'conv' in [XXX]	 #
# conventional time scale parameter 'phi'  in [1/msec]	 #
##########################################################
c       = 1
conv    = 9.55589e-2
phi     = 3

V_DOT   = -1. / c * (INA + IK + ICL_l)
N_DOT   =  phi * (AN * (1 - n) - BN * n)
H_DOT   =  phi * (AH * (1 - h) - BH * h)

KI_DOT  = -CONV/vli * IK
CLI_DOT =  CONV/vli * ICL_l

####################################################
# auxiliary variables for data output and plotting #
####################################################
aux _ki	   = ki
aux _ke	   = KE
aux _nai   = NAI
aux _nae   = NAE
aux _cli   = cli
aux _cle   = CLE

aux _EK	   = EK
aux _ENA   = ENA
aux _ECL   = ECL


####################################
# Numerical parameters: simulation #
####################################
@ meth=stiff
@ dt=5e-3
@ maxstor=10000000, bounds=10000000
@ total=200
@ bell=0

#############################################################
# Parameters for fixed point continuation in AUTO interface #
#############################################################
@ NTST=50, NMAX=115000, NPR=115000
@ DS=0.002, DSMIN=0.001, DSMAX=0.005
@ PARMIN=-250, PARMAX=50
@ AUTOXMIN=-250, AUTOXMAX=50, AUTOYMIN=-150, AUTOYMAX=50
@ EPSL=0.001, EPSU=0.001, EPSS=0.001


#################
# Plot settings #
###########################################################
# '_MAX_P' as a guide to the eye to see pump interruption #
###########################################################
@ xhi=200
@ nplot=3, yp1=v, yp2=_EK, yp3=_ENA, ylo=-150, yhi=160

done