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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,97 @@ | ||
| Cable Equation | ||
| ============== | ||
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| .. note:: | ||
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| NEURON has a sophisticated system for allowing users to describe the | ||
| geometry of neurons. Here we'll try to derive the equations in a manner that | ||
| hides those details whenever they're not relevant to NMODL. Please consult | ||
| its geometry related documentation or one of its publications, e.g. `The | ||
| NEURON Simulation Environment`_. | ||
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| .. _The NEURON Simulation Environment: https://doi.org/10.1162/neco.1997.9.6.1179 | ||
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| In order to derive the `cable equations` we model a neuron as an electrical | ||
| circuit. We first pick points along the neuron at which we model the voltage. | ||
| We'll call them nodes and connect the nodes to form a graph. At every branch | ||
| point we place a node, see Figure 1. | ||
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| .. figure:: ../images/cable-eqn_nodes.svg | ||
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| Figure 1: Illustration of the placement of node along a neurite. | ||
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| Two adjacent nodes are connected by a resistor. The interesting behaviour comes | ||
| from a difference in ion concentrations across the membrane. This difference is | ||
| upheld by three processes: a) the membrane which is largely impermeable to | ||
| ions, effectively creating a barrier for ions; b) (voltage-gated) ion channels | ||
| that conditionally allow ions to quickly cross the membrane; and c) ion pumps | ||
| which continuously pump ions across the membrane to restore a resting state. | ||
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| The fact that the membrane is (mostly) impermeable to ions means that it | ||
| behaves like a dielectric material and can therefore be modeled by a capacitor. | ||
| The ion pumps and channels we simply model by a current :math:`I`. | ||
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| This model gives rise to the circuit shown in Figure 2. | ||
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| .. figure:: ../images/cable-eqn_circuit.svg | ||
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| Figure 2: Illustration of the circuit near one node. The total trans-membrane | ||
| current is :math:`I_M`, the current due to the dielectric property of the | ||
| membrane is :math:`I_C`, and all mechanism specific currents are represented | ||
| by :math:`I`. | ||
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| We can start writing down equations. Let's recall the formula for a capacitor | ||
| and Ohm's Law: | ||
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| .. math:: | ||
| I = C \frac{dV}{dt}, \qquad | ||
| \Delta V = R I | ||
| Using Kirchoff's Law we can write down two equations for the trans-membrane | ||
| current: | ||
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| .. math:: | ||
| I_M &= I_C + I(V_1) \\ | ||
| I_{0,1} &= I_{1, 2} + I_M | ||
| which leads to | ||
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| .. math:: | ||
| I_C + I = I_{0,1} - I_{1, 2} \\ | ||
| I_C + I_{1,2} - I_{0, 1} = -I | ||
| which can be rewritten in terms of the voltage as follows: | ||
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| .. math:: | ||
| C \frac{dV_1}{dt} + R_{1,2}^{-1} (V_{2} - V{1}) - R_{0,1}^{-1} (V_{1} - V_{0}) = - I(V_1) | ||
| This can be discretized by implicit Euler: | ||
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| .. math:: | ||
| C \frac{V_1^{n+1} - V_1^{n}}{\Delta t} + R_{1,2}^{-1} \left(V_{2}^{n+1} - V_{1}^{n+1}\right) - R_{0,1}^{-1} \left(V_{1}^{n+1} - V_{0}^{n+1}\right) = - I(V_1^{n+1}) | ||
| We collect terms as follows: | ||
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| .. math:: | ||
| R_{0,1}^{-1} V_{0}^{n+1} | ||
| + \left(\frac{C}{\Delta t} + R_{0,1}^{-1} - R_{1,2}^{-1}\right) V_1^{n+1} | ||
| + R_{1,2}^{-1} V_{2}^{n+1} | ||
| = \frac{C}{\Delta t} V_1^{n} - I(V_1^{n+1}) | ||
| The unpleasant term is :math:`I(V_1^{n+1})` since it makes the system non-linear. | ||
| Therefore, it's linearized as follows: | ||
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| .. math:: | ||
| I(V_1^{n+1}) | ||
| &\approx I_1^{n} + \left(V^{n+1} - V^{n}\right) \frac{dI_1}{dV_1} \\ | ||
| &=: I_1^{n} + \left(V^{n+1} - V^{n}\right) g_i^{n} | ||
| where :math:`g_i^{n}` is the mechanism dependent (differential) conductance. | ||
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| Original file line number | Diff line number | Diff line change |
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| @@ -1,15 +1,44 @@ | ||
| #!/usr/bin/env bash | ||
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| # script for generating documentation for NMODL | ||
| # note that the NMODL Python wheel must be installed | ||
| # for the script to work properly | ||
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| set -xe | ||
| set -eu | ||
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| if [ $# -lt 1 ] | ||
| then | ||
| echo "Usage: $(basename "$0") output_dir [python_exe]" | ||
| exit 1 | ||
| fi | ||
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| # the dir where we put the temporary build and the docs | ||
| output_dir="$1" | ||
| # path to the Python executable | ||
| python_exe="${2:-"$(command -v python3)"}" | ||
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| if ! [ -d "${output_dir}" ] | ||
| then | ||
| mkdir -p "${output_dir}" | ||
| fi | ||
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| build_dir="build" | ||
| docs_dir="docs" | ||
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| echo "== Building documentation files in: ${output_dir}/${docs_dir} ==" | ||
| echo "== Temporary project build directory is: ${output_dir}/${build_dir} ==" | ||
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| venv_name="${output_dir}/env" | ||
| ${python_exe} -m venv "${venv_name}" | ||
| . "${venv_name}/bin/activate" | ||
| python_exe="$(command -v python)" | ||
| ${python_exe} -m pip install -U pip | ||
| ${python_exe} -m pip install ".[docs]" -C build-dir="${output_dir}/${build_dir}" | ||
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| # the abs dir where this script is located (so we can call it from wherever) | ||
| script_dir="$(realpath "$(dirname "$0")")" | ||
| script_dir="$(cd "$(dirname "$0")"; pwd -P)" | ||
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| cd "${script_dir}" | ||
| doxygen Doxyfile | ||
| sphinx-build . "${script_dir}/../public" | ||
| cd - | ||
| sphinx-build docs/ "${output_dir}/${docs_dir}" | ||
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| deactivate |
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