opensourcebrain/HodgkinHuxleyTutorial

1.. role:: raw-html(raw)
2   :format: html
3
4Biological/Electronic Equivalence
5=================================
6
7This model relies on a basic equivalence between a biological membrane plus
8embedded ion channels, and an electronic circuit.
9
10The circuit can be described by the diagram below, which is an electronic
11diagram representing a patch of cellular membrane.
12
13.. image:: ../_media/equivalentCircuit.png
14
15
16The Circuit
17-----------
18
19The **membrane capacitance** ( |Cm| ) is taken to be a fixed property of the membrane.
20
21
22Parallel to |Cm| are two "battery-capacitor" series; one for each of
23voltage-gated and leak ion channels.
24
25Each of these ion pathways are modeled as the product of the ion's
26**conductance** ( g ) and its driving electrochemical gradient ( E ), both of which may vary
27over time (except in the case of |gL| ; see below).
28
29|Ip| represents the active movement of ions provided by
30**ion transporters**.
31
32The net result of all of this activity in the cell membrane is a current across
33the membrane (i.e. from intracellular medium to extracellular medium, or vice versa).
34
35Modeling voltage-gating versus leak:
36^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
37
38The conductances of voltage-gated and leak channels, |gn| and |gL|
39respectively, are modeled differently. Since the gating of voltage-gated ion
40channels depends on the membrane potential at a given moment, it is non-linear.
41In contrast, leak ion channels are always in the same state, so their
42conductance is modeled linearly.
43
44The Math
45--------
46
47Lipid bilayer current
48^^^^^^^^^^^^^^^^^^^^^
49
50.. image:: http://upload.wikimedia.org/math/2/2/4/224f520989592dc0d3aa096313581e19.png
51
52The current across the cell's lipid bilayer ( |Ic| ) is the product of the
53membrane's capacitance ( |Cm| ) and the rate of change of membrane
54potential ( |Vm| ) with respect to time ( t ).
55
56Ion channel current
57^^^^^^^^^^^^^^^^^^^
58
59.. image:: http://upload.wikimedia.org/math/6/1/7/617b32943eae50e0e9f34cc5d0f4faf4.png
60
61The current through a given ion channel ( |Ii| ) is the product of that
62channel's conductance ( |gi| ) and the difference |Vm| - |Vi|
63
64|Vi| is the ion species' **reversal potential**. Notice that when |Vm|
65is equal to |Vi| the product becomes zero, and there is no net flow
66( |Ii| ) for the ion, which is what defines reversal potential.
67
68Combining these currents
69^^^^^^^^^^^^^^^^^^^^^^^^
70
71.. image:: http://upload.wikimedia.org/math/0/a/4/0a40dd385ad9546d4722cccacd11120b.png
72
73If we sum the lipid bilayer current with ion channel currents for each ion
74species, we end up with a total current ( I ) for the patch of cellular
75membrane.
76
77In the equation above, voltage-gated potassium ( K ) and sodium ( Na ) channels,
78as well as leak channels ( L ) are considered, leading to three instances of
79the ion channel current calculations.
80
81Adding in activation parameters
82^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
83
84Since the ion channels denoted in the equation above are in various states, some
85new variables must be added to the equation. Namely, activation and inactivation
86parameters are now included in each ion channel current calculation.
87
88.. image:: https://upload.wikimedia.org/math/e/2/6/e26962e13109f3e6df273553a731f24b.png
89
90The new notation for each conductance variable (|g_|)
91is the *maximal* conductance for that ion channel type. This, combined with the
92activation/inactivation parameters *n*, *m* and *h*, still represents the level
93of conductance for an ion channel, but with parameters that modify this
94conductance.
95
96- *m*  is the activation parameter for sodium ( Na ) channels
97- *h* is the inactivation parameter for sodium channels
98- *n* is the activation parameter for potassium ( K ) channels
99
100Plots
101-----
102
103The plots generated by running the `HodgkinHuxley.py script <Hodgkin%20Huxley.html>`_
104bundled with this tutorial are show below.
105
106.. image:: ../_media/figure_1.png
107
108Description of each plot
109^^^^^^^^^^^^^^^^^^^^^^^^
110
111Starting from the bottom, the first (bottom-most) plot shows neural membrane voltage activity.
112The spikes here are called "action potentials" and correspond directly to the
113current/time plot. Outflux of *Na* directly followed by influx of *K* causes the spiking activity 
114observed in the plot.
115
116The second plot from the bottom (let's call this the *gating plot*) shows the activation/inactivation parameters of
117the ion channels in the neuron. The precise meanings of the three lines labeled
118*m*, *h* and *n* are described above, but it is sufficient to say that these
119parameters are proportional to the "amount" of gating for their respective ion
120channels. In other words, the amount of influence of each parameter on internal
121dynamics is given proportional to its full possible influence. So a gating
122value of 1 is at its maximal influence, and zero is no influence at all.
123
124The third plot from the bottom (the *current/time plot*) makes this more concrete, showing the influx
125(negative y-axis) and outflux (positive y-axis) of ions passing through each
126type of ion channel being modeled. Notice that, in the gating plot, at times
127when *m* is large and *h* is small for a moment (say, just after 100ms), the
128sodium current (|Ina|) spikes *outward*. Notice also that the potassium current
129(|Ik|) spikes inward when its activation parameter *n* spikes in the gating
130plot.
131
132Finally, consider the top plot, which shows two currents injected into the
133cell membrane at times 100ms and 300ms. Notice that the second injected current
134is significantly larger in magnitude than the first.
135
136
137Relationship between the plots
138^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
139
140Notice that the first set of action potentials (from about 100 to 200ms) is
141sparse compared to the second set (from 300 to 400ms). This is due to the
142increased current applied across the membrane in the second injection (see the
143bottom plot).
144
145It is possible to see how intracellular and cell-patch dynamics are related
146through these four plots. Gating parameters affect ion channel conductance,
147which directly influences ion flow, which in turn controls electric potential
148across the membrane.
149
150Terms
151-----
152
153- `Ion channel <http://en.wikipedia.org/wiki/Ion_channel>`_
154    - Protein embedded in cellular membrane allowing *passive* flow of ions, depending on its configuration.
155- Ion channel conductance
156    - The rate of flow of ions through an ion channel. Directly affects membrane conductance, and changes with gating behaviour of an ion channel.
157- `Ion transporter <http://en.wikipedia.org/wiki/Ion_transporter>`_
158    - Protein embedded in cellular membrane that moves ions *actively*
159- `Membrane capacitance <http://www.scholarpedia.org/article/Electrical_properties_of_cell_membranes#Capacitance>`_
160- `Membrane conductance <http://www.scholarpedia.org/article/Electrical_properties_of_cell_membranes#Conductance>`_
161    - Total membrane conductance is the rate at which current (i.e. ions) can flow through the membrane, and is a result of the configuration of ion channels at a given moment.
162- `Membrane potential <https://en.wikipedia.org/wiki/Membrane_potential>`_
163    - The difference in electric potential between the exterior and interior of a cell.
164- Nernst potential
165    - See "Reversal potential".
166- `Reversal potential <https://en.wikipedia.org/wiki/Reversal_potential>`_
167    - The membrane potential at which a given ion species has no overall flow across the membrane (i.e. the ion flow direction "reverses").
168
169.. |Cm| replace:: C\ :sub:`m`
170.. |g_| replace:: :raw-html:`<span style="text-decoration:overline">g</span>`
171.. |gi| replace:: g\ :sub:`i`
172.. |gL| replace:: g\ :sub:`L`
173.. |gn| replace:: g\ :sub:`n`
174.. |Ic| replace:: I\ :sub:`c`
175.. |Ii| replace:: I\ :sub:`i`
176.. |Ik| replace:: I\ :sub:`k`
177.. |Ina| replace:: I\ :sub:`na`
178.. |Ip| replace:: I\ :sub:`p`
179.. |Vi| replace:: V\ :sub:`i`
180.. |Vm| replace:: V\ :sub:`m`
181