The Hodgkin-Huxley Circuit for a Neuron
James Sochacki & Jeffrey Kopsick
James Madison University - George Mason University
March 26, 2021
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 1 / 31 Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 Alex Gofen - http://taylorcenter.org/Gofen/
Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 Extending Power Series Methods for the Hodgkin-Huxley Equations Including Sensitive Dependence Acknowledgements
G. Edgar Parker Robert D. Stewart & Wyeth Bair JMU Power Series Team - Carothers, Lucas, Thelwell, Tongen, Warne, D. & Warne, P. - http://educ.jmu.edu/~sochacjs/PSM.html James Money - DOE Thomas Szewczyk - DOE Anthony England - DOE Intern Adam Miller - DOE Intern Richard Neidinger - Davidson Alex Gofen - http://taylorcenter.org/Gofen/
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 2 / 31 James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 3 / 31 120
100
80
60
40
Voltage - Millivolts 20
0
-20
-40 0 10 20 30 40 50 60 70 80 TIME - Milliseconds
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 4 / 31 James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 5 / 31 d n = α (V ) (1 − n) − β (V ) n dt n n d m = α (V ) (1 − m) − β (V ) m dt m m d h = α (V ) (1 − h) − β (V ) dt h h
The original Hodgkin-Huxley ODEs
0 4 3 IM = CM V +g ¯K n (V − EK ) +g ¯Na m h (V − ENa) +g ¯L (V − EL)
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 6 / 31 d m = α (V ) (1 − m) − β (V ) m dt m m d h = α (V ) (1 − h) − β (V ) dt h h
The original Hodgkin-Huxley ODEs
0 4 3 IM = CM V +g ¯K n (V − EK ) +g ¯Na m h (V − ENa) +g ¯L (V − EL) d n = α (V ) (1 − n) − β (V ) n dt n n
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 6 / 31 d h = α (V ) (1 − h) − β (V ) dt h h
The original Hodgkin-Huxley ODEs
0 4 3 IM = CM V +g ¯K n (V − EK ) +g ¯Na m h (V − ENa) +g ¯L (V − EL) d n = α (V ) (1 − n) − β (V ) n dt n n d m = α (V ) (1 − m) − β (V ) m dt m m
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 6 / 31 The original Hodgkin-Huxley ODEs
0 4 3 IM = CM V +g ¯K n (V − EK ) +g ¯Na m h (V − ENa) +g ¯L (V − EL) d n = α (V ) (1 − n) − β (V ) n dt n n d m = α (V ) (1 − m) − β (V ) m dt m m d h = α (V ) (1 − h) − β (V ) dt h h
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 6 / 31 The original Hodgkin-Huxley ODEs
0 4 3 IM = CM V +g ¯K n (V − EK ) +g ¯Na m h (V − ENa) +g ¯L (V − EL) d n = α (V ) (1 − n) − β (V ) n dt n n d m = α (V ) (1 − m) − β (V ) m dt m m d h = α (V ) (1 − h) − β (V ) dt h h
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 6 / 31 0 n = − (αn(V ) + βn(V )) n + αn(V ) 0 m = − (αm(V ) + βm(V )) m + αm(V ) 0 h = − (αh(V ) + βh(V )) h + αh(V )
The HH ODEs can be written as
0 1 4 3 V = − g¯K n(t) +g ¯Nam(t) h(t) +g ¯L V + CM 1 4 3 IM (t) g¯K EK n(t) +g ¯NaENam(t) h(t) +g ¯LEL − CM CM
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 7 / 31 The HH ODEs can be written as
0 1 4 3 V = − g¯K n(t) +g ¯Nam(t) h(t) +g ¯L V + CM 1 4 3 IM (t) g¯K EK n(t) +g ¯NaENam(t) h(t) +g ¯LEL − CM CM 0 n = − (αn(V ) + βn(V )) n + αn(V ) 0 m = − (αm(V ) + βm(V )) m + αm(V ) 0 h = − (αh(V ) + βh(V )) h + αh(V )
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 7 / 31 αn(Vrest ) nE = (αn(Vrest + βn(Vrest ))
αm(Vrest ) mE = (αm(Vrest ) + βm(Vrest )) αh(Vrest ) hE = (αh(Vrest ) + βh(Vrest ))
be the equilibrium values for n, m, h, respectively then
4 3 g¯K nE EK +g ¯Na mE hE ENa +g ¯LEL Vrest = 4 3 g¯K nE +g ¯Na mE hE +g ¯L
If one lets
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 8 / 31 be the equilibrium values for n, m, h, respectively then
4 3 g¯K nE EK +g ¯Na mE hE ENa +g ¯LEL Vrest = 4 3 g¯K nE +g ¯Na mE hE +g ¯L
If one lets
αn(Vrest ) nE = (αn(Vrest + βn(Vrest ))
αm(Vrest ) mE = (αm(Vrest ) + βm(Vrest )) αh(Vrest ) hE = (αh(Vrest ) + βh(Vrest ))
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 8 / 31 4 3 g¯K nE EK +g ¯Na mE hE ENa +g ¯LEL Vrest = 4 3 g¯K nE +g ¯Na mE hE +g ¯L
If one lets
αn(Vrest ) nE = (αn(Vrest + βn(Vrest ))
αm(Vrest ) mE = (αm(Vrest ) + βm(Vrest )) αh(Vrest ) hE = (αh(Vrest ) + βh(Vrest ))
be the equilibrium values for n, m, h, respectively then
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 8 / 31 If one lets
αn(Vrest ) nE = (αn(Vrest + βn(Vrest ))
αm(Vrest ) mE = (αm(Vrest ) + βm(Vrest )) αh(Vrest ) hE = (αh(Vrest ) + βh(Vrest ))
be the equilibrium values for n, m, h, respectively then
4 3 g¯K nE EK +g ¯Na mE hE ENa +g ¯LEL Vrest = 4 3 g¯K nE +g ¯Na mE hE +g ¯L
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 8 / 31 0 k−1 k V (t) = V1 + 2V2t + ... + kVk t + (k + 1)Vk+1t k X i = (i + 1)Vi+1t i=0
(k + 1)Vk+1 = P1(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)nk+1 = P2(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)mk+1 = P3(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)hk+1 = P4(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
k 2 k X i V (t) = V0 + V1t + V2t + ... + Vk t = Vi t i=0
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 9 / 31 (k + 1)Vk+1 = P1(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)nk+1 = P2(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)mk+1 = P3(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)hk+1 = P4(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
k 2 k X i V (t) = V0 + V1t + V2t + ... + Vk t = Vi t i=0 0 k−1 k V (t) = V1 + 2V2t + ... + kVk t + (k + 1)Vk+1t k X i = (i + 1)Vi+1t i=0
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 9 / 31 (k + 1)nk+1 = P2(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)mk+1 = P3(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)hk+1 = P4(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
k 2 k X i V (t) = V0 + V1t + V2t + ... + Vk t = Vi t i=0 0 k−1 k V (t) = V1 + 2V2t + ... + kVk t + (k + 1)Vk+1t k X i = (i + 1)Vi+1t i=0
(k + 1)Vk+1 = P1(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 9 / 31 k 2 k X i V (t) = V0 + V1t + V2t + ... + Vk t = Vi t i=0 0 k−1 k V (t) = V1 + 2V2t + ... + kVk t + (k + 1)Vk+1t k X i = (i + 1)Vi+1t i=0
(k + 1)Vk+1 = P1(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)nk+1 = P2(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)mk+1 = P3(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
(k + 1)hk+1 = P4(V0, ..., Vk , n0, ..., nk , m0, ..., mk , h0, ..., hk )
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 9 / 31 Hodgkin and Huxley experimentally determined αn, βn, αm, βm, αh, βh for a particular squid to be
0.01(10−V ) V αn = 10−V ; βn = 0.125 exp 80 exp −1 10
0.1(25−V ) −V αm = 25−V ; βm = 4 exp 18 exp −1 10
−V 1 αh = 0.07 exp 20 ; βh = 30−V . exp +1 10
The HH Parameters & The Removable Singularities
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 10 / 31 0.01(10−V ) V αn = 10−V ; βn = 0.125 exp 80 exp −1 10
0.1(25−V ) −V αm = 25−V ; βm = 4 exp 18 exp −1 10
−V 1 αh = 0.07 exp 20 ; βh = 30−V . exp +1 10
The HH Parameters & The Removable Singularities
Hodgkin and Huxley experimentally determined αn, βn, αm, βm, αh, βh for a particular squid to be
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 10 / 31 0.1(25−V ) −V αm = 25−V ; βm = 4 exp 18 exp −1 10
−V 1 αh = 0.07 exp 20 ; βh = 30−V . exp +1 10
The HH Parameters & The Removable Singularities
Hodgkin and Huxley experimentally determined αn, βn, αm, βm, αh, βh for a particular squid to be
0.01(10−V ) V αn = 10−V ; βn = 0.125 exp 80 exp −1 10
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 10 / 31 −V 1 αh = 0.07 exp 20 ; βh = 30−V . exp +1 10
The HH Parameters & The Removable Singularities
Hodgkin and Huxley experimentally determined αn, βn, αm, βm, αh, βh for a particular squid to be
0.01(10−V ) V αn = 10−V ; βn = 0.125 exp 80 exp −1 10
0.1(25−V ) −V αm = 25−V ; βm = 4 exp 18 exp −1 10
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 10 / 31 The HH Parameters & The Removable Singularities
Hodgkin and Huxley experimentally determined αn, βn, αm, βm, αh, βh for a particular squid to be
0.01(10−V ) V αn = 10−V ; βn = 0.125 exp 80 exp −1 10
0.1(25−V ) −V αm = 25−V ; βm = 4 exp 18 exp −1 10
−V 1 αh = 0.07 exp 20 ; βh = 30−V . exp +1 10
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 10 / 31 x 0(ex − 1) − xx 0ex ex (1 − x) − 1 α0 = 0.1 = 0.1x 0 . n (ex − 1)2 (ex − 1)2
x −1 x Let αn,1 = (e − 1) and αn,2 = e .
0 x −2 0 2 0 αn,1 = −(e − 1) x = −αn,1 x
0 x 0 0 αn,2 = e x = αn,2x .
Substituting these back into the ODE for αn leads to
x α = 0.1 n exp (x) − 1
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 11 / 31 x −1 x Let αn,1 = (e − 1) and αn,2 = e .
0 x −2 0 2 0 αn,1 = −(e − 1) x = −αn,1 x
0 x 0 0 αn,2 = e x = αn,2x .
Substituting these back into the ODE for αn leads to
x α = 0.1 n exp (x) − 1
x 0(ex − 1) − xx 0ex ex (1 − x) − 1 α0 = 0.1 = 0.1x 0 . n (ex − 1)2 (ex − 1)2
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 11 / 31 0 x 0 0 αn,2 = e x = αn,2x .
Substituting these back into the ODE for αn leads to
x α = 0.1 n exp (x) − 1
x 0(ex − 1) − xx 0ex ex (1 − x) − 1 α0 = 0.1 = 0.1x 0 . n (ex − 1)2 (ex − 1)2
x −1 x Let αn,1 = (e − 1) and αn,2 = e .
0 x −2 0 2 0 αn,1 = −(e − 1) x = −αn,1 x
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 11 / 31 Substituting these back into the ODE for αn leads to
x α = 0.1 n exp (x) − 1
x 0(ex − 1) − xx 0ex ex (1 − x) − 1 α0 = 0.1 = 0.1x 0 . n (ex − 1)2 (ex − 1)2
x −1 x Let αn,1 = (e − 1) and αn,2 = e .
0 x −2 0 2 0 αn,1 = −(e − 1) x = −αn,1 x
0 x 0 0 αn,2 = e x = αn,2x .
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 11 / 31 x α = 0.1 n exp (x) − 1
x 0(ex − 1) − xx 0ex ex (1 − x) − 1 α0 = 0.1 = 0.1x 0 . n (ex − 1)2 (ex − 1)2
x −1 x Let αn,1 = (e − 1) and αn,2 = e .
0 x −2 0 2 0 αn,1 = −(e − 1) x = −αn,1 x
0 x 0 0 αn,2 = e x = αn,2x .
Substituting these back into the ODE for αn leads to James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 11 / 31 0 0 2 = 0.1(x αn,1 − xx αn,1 ).
Therefore, as long as x 6= 0 and y 6= 0, one can create polynomial ODEs for αn and αm.
x 0 α0 = 0.1 = 0.1(xα )0 = 0.1(x 0α + xα0 ) n exp (x) − 1 n,1 n,1 n,1
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 12 / 31 Therefore, as long as x 6= 0 and y 6= 0, one can create polynomial ODEs for αn and αm.
x 0 α0 = 0.1 = 0.1(xα )0 = 0.1(x 0α + xα0 ) n exp (x) − 1 n,1 n,1 n,1 0 0 2 = 0.1(x αn,1 − xx αn,1 ).
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 12 / 31 x 0 α0 = 0.1 = 0.1(xα )0 = 0.1(x 0α + xα0 ) n exp (x) − 1 n,1 n,1 n,1 0 0 2 = 0.1(x αn,1 − xx αn,1 ).
Therefore, as long as x 6= 0 and y 6= 0, one can create polynomial ODEs for αn and αm.
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 12 / 31 Best Fit:
αn2,m2 = a ln(exp(cx) + r) + bx,
LN:
αn3 = 0.1 (ln (exp(x) + 1) − x)
αm3 = ln (exp(y) + 1) − y
EXP:
αn4,m4 = p exp(sx) + q
Three Approximations for αn and αm
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 13 / 31 LN:
αn3 = 0.1 (ln (exp(x) + 1) − x)
αm3 = ln (exp(y) + 1) − y
EXP:
αn4,m4 = p exp(sx) + q
Three Approximations for αn and αm Best Fit:
αn2,m2 = a ln(exp(cx) + r) + bx,
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 13 / 31 EXP:
αn4,m4 = p exp(sx) + q
Three Approximations for αn and αm Best Fit:
αn2,m2 = a ln(exp(cx) + r) + bx,
LN:
αn3 = 0.1 (ln (exp(x) + 1) − x)
αm3 = ln (exp(y) + 1) − y
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 13 / 31 Three Approximations for αn and αm Best Fit:
αn2,m2 = a ln(exp(cx) + r) + bx,
LN:
αn3 = 0.1 (ln (exp(x) + 1) − x)
αm3 = ln (exp(y) + 1) − y
EXP:
αn4,m4 = p exp(sx) + q
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 13 / 31 Plot of All Four Plot of All Four n m 2.5 30
2
1.5 20
1 10 0.5
0 0 -30 0 30 60 90 120 -30 0 30 60 90 120
Plot of , , Plot of , , n n n m m m 2 3 2 3 10
1 8
6 0.5 4
2 0 0 -30 0 30 60 90 120 -30 0 30 60 90 120
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 14 / 31 Table: Original Hodgkin-Huxley Parameter Equilibria
α Vrest nE mE hE HH 0.003621 0.317732 0.052955 0.595994 BF 0.003617 0.317732 0.052955 0.595994 LN 3.317822 0.256558 0.031529 0.477551 EXP 0.004388 0.317720 0.052954 0.595967
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 15 / 31 ex − 1 g(x) = , x 6= 0 x
Assume
g(0) = lim ex −1 = 1 x→0 x
∞ 1 1 X x n g(x) = 1 + x + x 2 + ... = 2 3! (n + 1)! n=0 Question: What is g(V (t))?
Addressing the Removable Singularity
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 16 / 31 Assume
g(0) = lim ex −1 = 1 x→0 x
∞ 1 1 X x n g(x) = 1 + x + x 2 + ... = 2 3! (n + 1)! n=0 Question: What is g(V (t))?
Addressing the Removable Singularity ex − 1 g(x) = , x 6= 0 x
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 16 / 31 ∞ 1 1 X x n g(x) = 1 + x + x 2 + ... = 2 3! (n + 1)! n=0 Question: What is g(V (t))?
Addressing the Removable Singularity ex − 1 g(x) = , x 6= 0 x
Assume
g(0) = lim ex −1 = 1 x→0 x
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 16 / 31 Question: What is g(V (t))?
Addressing the Removable Singularity ex − 1 g(x) = , x 6= 0 x
Assume
g(0) = lim ex −1 = 1 x→0 x
∞ 1 1 X x n g(x) = 1 + x + x 2 + ... = 2 3! (n + 1)! n=0
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 16 / 31 Addressing the Removable Singularity ex − 1 g(x) = , x 6= 0 x
Assume
g(0) = lim ex −1 = 1 x→0 x
∞ 1 1 X x n g(x) = 1 + x + x 2 + ... = 2 3! (n + 1)! n=0 Question: What is g(V (t))?
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 16 / 31 th Let pM be the M degree (Maclaurin) polynomial defined by
M X m 2 M pM = pM,mt = pM,0 + pM,1 t + pM,2 t + ... + pM,M t m=0
th and qN be the N degree (Maclaurin) polynomial defined by
N X n 2 N qN = qN,nt = qN,0 + qN,1 t + qN,2 t + ... + qN,N t n=0
then the (Maclaurin) polynomial for pM (qN (t)) can be obtained by letting
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 17 / 31 M X pM (qN ) = pM,mwm = pM,0 w0 + pM,1 w1 + pM,2 w2 + ... + pM,M wM m=0
w0 = 1
w1 = qN 2 w2 = (qN ) . . . M wM = (qN )
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 18 / 31 w0 = 1
w1 = qN 2 w2 = (qN ) . . . M wM = (qN )
M X pM (qN ) = pM,mwm = pM,0 w0 + pM,1 w1 + pM,2 w2 + ... + pM,M wM m=0
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 18 / 31 J X j wm(t) = wm,j t , m = 0..M j=0
0 w0 = 0 0 0 w1 = qN 0 0 w2 = 2w1 w1 . . . 0 0 wM = MwM−1 w1
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 19 / 31 0 w0 = 0 0 0 w1 = qN 0 0 w2 = 2w1 w1 . . . 0 0 wM = MwM−1 w1
J X j wm(t) = wm,j t , m = 0..M j=0
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 19 / 31 M J ! X X j pM (qN (t)) = pM,m wm,j t m=0 j=0 M J M ! X X X j = pM,mwm,0 + pM,mwm,j t m=0 j=1 m=1
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 20 / 31 [V ] 2 3 4 M8 (t) = 0 + 109t − 613t + 4131t − 28, 299t + 203, 296t5 − 1, 511, 197t6 + 11, 548, 619t7 − 90, 113, 613t8
The ratios of these Maclaurin coefficients are
{−5.6, −6.74, −6.85, −7.18, −7.43, −7.64, −7.8, −7.93}.
Maclaurin Coefficients
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 21 / 31 The ratios of these Maclaurin coefficients are
{−5.6, −6.74, −6.85, −7.18, −7.43, −7.64, −7.8, −7.93}.
Maclaurin Coefficients
[V ] 2 3 4 M8 (t) = 0 + 109t − 613t + 4131t − 28, 299t + 203, 296t5 − 1, 511, 197t6 + 11, 548, 619t7 − 90, 113, 613t8
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 21 / 31 {−5.6, −6.74, −6.85, −7.18, −7.43, −7.64, −7.8, −7.93}.
Maclaurin Coefficients
[V ] 2 3 4 M8 (t) = 0 + 109t − 613t + 4131t − 28, 299t + 203, 296t5 − 1, 511, 197t6 + 11, 548, 619t7 − 90, 113, 613t8
The ratios of these Maclaurin coefficients are
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 21 / 31 Maclaurin Coefficients
[V ] 2 3 4 M8 (t) = 0 + 109t − 613t + 4131t − 28, 299t + 203, 296t5 − 1, 511, 197t6 + 11, 548, 619t7 − 90, 113, 613t8
The ratios of these Maclaurin coefficients are
{−5.6, −6.74, −6.85, −7.18, −7.43, −7.64, −7.8, −7.93}.
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 21 / 31 120
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James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 22 / 31 James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 23 / 31 120 120 120
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James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 24 / 31 120 120 Voltage Spike 100 100
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James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 25 / 31 James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 26 / 31 James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 27 / 31 James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 28 / 31 James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 29 / 31 ∂V C = F (V (x + ∆x) − V (x, t)) + F (V (x) − V (x − ∆x, t)) M ∂t + − 4 3 − (¯gK n(x, t) (V − EK ) +g ¯Na m(x, t) h(x, t)(V − ENa)
+g ¯L (V (x, t) − EL) + IM (x, t)) 0 n (x, t) = − (αn(V (x, t)) + βn(V (x, t))) n(x, t) + αn(V (x, t)) 0 m (x, t) = − (αm(V (x, t)) + βm(V (x, t))) m(x, t) + αm(V (x, t)) 0 h (x, t) = − (αh(V (x, t)) + βh(V (x, t))) h(x, t) + αh(V (x, t))
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 30 / 31 ANIMATIONS
James Sochacki & Jeffrey Kopsick (James MadisonTheHodgkin-Huxley University - George Circuit Mason for University)a Neuron March 26, 2021 31 / 31