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)