{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Initialization Strategies"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A version of this notebook may be run online via Google Colab at https://tinyurl.com/rxd-basic-initialization (make a copy or open in playground mode)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The time series of a chemical concentration necessarily depends on its initial conditions; i.e. the concentration at time 0. An analogous statement is true for gating variables, etc. How do we specify this?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Option 1: NEURON and NMODL defaults"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If the species corresponds to one with initial conditions specified by NMODL (or in the case of sodium, potassium, or calcium with meaningful NEURON defaults), then omitting the initial argument will tell NEURON to use those rules. e.g."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.301606Z",
"iopub.status.busy": "2025-08-18T03:36:17.301455Z",
"iopub.status.idle": "2025-08-18T03:36:17.718524Z",
"shell.execute_reply": "2025-08-18T03:36:17.718165Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ca: 5e-05 mM\n",
"na: 10.0 mM\n",
"k: 54.4 mM\n",
"unknown: 1.0 mM\n"
]
}
],
"source": [
"from neuron import n, rxd\n",
"from neuron.units import mV\n",
"\n",
"soma = n.Section(name=\"soma\")\n",
"cyt = rxd.Region(soma.wholetree(), name=\"cyt\", nrn_region=\"i\")\n",
"\n",
"ca = rxd.Species(cyt, name=\"ca\", charge=2, atolscale=1e-6)\n",
"na = rxd.Species(cyt, name=\"na\", charge=1)\n",
"k = rxd.Species(cyt, name=\"k\", charge=1)\n",
"unknown = rxd.Species(cyt, name=\"unknown\", charge=-1)\n",
"\n",
"n.finitialize(-65 * mV)\n",
"\n",
"print(f\"ca: {ca.nodes[0].concentration} mM\")\n",
"print(f\"na: {na.nodes[0].concentration} mM\")\n",
"print(f\"k: {k.nodes[0].concentration} mM\")\n",
"print(f\"unknown: {unknown.nodes[0].concentration} mM\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As shown here, unknown ions/proteins are by default assigned a concentration by NEURON of 1 mM. The atolscale value for calcium has no effect on the initialized value, but is included here as an example of best practice for working with low concentrations."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Importantly, the NEURON/NMODL rules only apply if there is a corresponding classical NEURON state variable. That is, nrn_region must be set and the Species must have a name assigned."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Running what is otherwise the same code without the nrn_region assigned causes everything to default to 0 µM:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.780777Z",
"iopub.status.busy": "2025-08-18T03:36:17.780522Z",
"iopub.status.idle": "2025-08-18T03:36:17.795765Z",
"shell.execute_reply": "2025-08-18T03:36:17.795391Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ca: 0.0 mM\n",
"na: 0.0 mM\n",
"k: 0.0 mM\n",
"unknown: 0.0 mM\n"
]
}
],
"source": [
"from neuron import n, rxd\n",
"from neuron.units import mV\n",
"\n",
"soma = n.Section(name=\"soma\")\n",
"cyt = rxd.Region(soma.wholetree(), name=\"cyt\")\n",
"\n",
"ca = rxd.Species(cyt, name=\"ca\", charge=2)\n",
"na = rxd.Species(cyt, name=\"na\", charge=1)\n",
"k = rxd.Species(cyt, name=\"k\", charge=1)\n",
"unknown = rxd.Species(cyt, name=\"unknown\", charge=-1)\n",
"\n",
"n.finitialize(-65 * mV)\n",
"\n",
"print(f\"ca: {ca.nodes[0].concentration} mM\")\n",
"print(f\"na: {na.nodes[0].concentration} mM\")\n",
"print(f\"k: {k.nodes[0].concentration} mM\")\n",
"print(f\"unknown: {unknown.nodes[0].concentration} mM\")"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.798824Z",
"iopub.status.busy": "2025-08-18T03:36:17.798674Z",
"iopub.status.idle": "2025-08-18T03:36:17.803526Z",
"shell.execute_reply": "2025-08-18T03:36:17.803147Z"
}
},
"outputs": [],
"source": [
"## get rid of previous model\n",
"soma = ca = na = k = unknown = None"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For extracellular species, there is no equivalent traditional NEURON state variable (as those only exist within and along the cell), however NEURON's constant initialization parameters for the nrn_region='o' space are used if available; e.g."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.805191Z",
"iopub.status.busy": "2025-08-18T03:36:17.805051Z",
"iopub.status.idle": "2025-08-18T03:36:17.812775Z",
"shell.execute_reply": "2025-08-18T03:36:17.812412Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ca: 0.42 mM\n"
]
}
],
"source": [
"from neuron import n, rxd\n",
"from neuron.units import mV\n",
"\n",
"ecs = rxd.Extracellular(\n",
" -100, -100, -100, 100, 100, 100, dx=20, volume_fraction=0.2, tortuosity=1.6\n",
")\n",
"\n",
"## defining calcium on both intra- and extracellular regions\n",
"ca = rxd.Species(ecs, name=\"ca\", charge=2)\n",
"\n",
"## global initialization for NEURON extracellular calcium\n",
"n.cao0_ca_ion = 0.42\n",
"\n",
"n.finitialize(-65 * mV)\n",
"\n",
"print(f\"ca: {ca.nodes[0].concentration} mM\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We could do something similar using cai0_ca_ion to set the global initial intracellular calcium concentration."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.816802Z",
"iopub.status.busy": "2025-08-18T03:36:17.816657Z",
"iopub.status.idle": "2025-08-18T03:36:17.819431Z",
"shell.execute_reply": "2025-08-18T03:36:17.819059Z"
}
},
"outputs": [],
"source": [
"## get rid of previous model\n",
"soma = ca = ecs = None"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Option 2: Uniform initial concentration"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Setting initial= to a Species or State assigns that value every time the system reinitializes. e.g."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.821158Z",
"iopub.status.busy": "2025-08-18T03:36:17.821019Z",
"iopub.status.idle": "2025-08-18T03:36:17.828395Z",
"shell.execute_reply": "2025-08-18T03:36:17.827528Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"m = 0.47\n"
]
}
],
"source": [
"from neuron import n, rxd\n",
"from neuron.units import mV\n",
"\n",
"soma = n.Section(name=\"soma\")\n",
"\n",
"cyt = rxd.Region([soma], name=\"cyt\")\n",
"m = rxd.State(cyt, initial=0.47)\n",
"\n",
"n.finitialize(-65 * mV)\n",
"print(f\"m = {m.nodes[0].value}\")"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.831803Z",
"iopub.status.busy": "2025-08-18T03:36:17.831659Z",
"iopub.status.idle": "2025-08-18T03:36:17.834354Z",
"shell.execute_reply": "2025-08-18T03:36:17.833990Z"
}
},
"outputs": [],
"source": [
"## get rid of previous model\n",
"m = cyt = soma = None"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Option 3: Initializing to a function of position"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The initial= keyword argument also accepts a callable (e.g. a function) that receives a node object. Nodes have certain properties that are useful for assinging based on position, including .segment (intracellular nodes only) and .x3d, .y3d, and .z3d. Segment-to-segment (or the segment containing a node) distances can be measured directly using n.distance."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Using n.distance:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we use the morphology c91662.CNG.swc obtained from NeuroMorpho.Org and initialize based on path distance from the soma."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:17.837621Z",
"iopub.status.busy": "2025-08-18T03:36:17.837472Z",
"iopub.status.idle": "2025-08-18T03:36:18.060719Z",
"shell.execute_reply": "2025-08-18T03:36:18.060234Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"--2025-08-18 03:36:17-- https://raw.githubusercontent.com/neuronsimulator/resources/8b1290d5c8ab748dd6251be5bd46a4e3794d742f/notebooks/rxd/c91662.CNG.swc\r\n",
"Resolving raw.githubusercontent.com (raw.githubusercontent.com)... 185.199.109.133, 185.199.110.133, 185.199.108.133, ...\r\n",
"Connecting to raw.githubusercontent.com (raw.githubusercontent.com)|185.199.109.133|:443... connected.\r\n",
"HTTP request sent, awaiting response... "
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"200 OK\r\n",
"Length: 55074 (54K) [text/plain]\r\n",
"Saving to: ‘c91662.CNG.swc’\r\n",
"\r\n",
"\r",
"c91662.CNG.swc 0%[ ] 0 --.-KB/s \r",
"c91662.CNG.swc 100%[===================>] 53.78K --.-KB/s in 0.001s \r\n",
"\r\n",
"Last-modified header missing -- time-stamps turned off.\r\n",
"2025-08-18 03:36:17 (89.5 MB/s) - ‘c91662.CNG.swc’ saved [55074/55074]\r\n",
"\r\n"
]
}
],
"source": [
"!wget -N https://raw.githubusercontent.com/neuronsimulator/resources/8b1290d5c8ab748dd6251be5bd46a4e3794d742f/notebooks/rxd/c91662.CNG.swc"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:18.063109Z",
"iopub.status.busy": "2025-08-18T03:36:18.062940Z",
"iopub.status.idle": "2025-08-18T03:36:18.723810Z",
"shell.execute_reply": "2025-08-18T03:36:18.723403Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"1.0"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from neuron import n, gui, rxd\n",
"from neuron.units import mV\n",
"\n",
"n.load_file(\"stdrun.hoc\")\n",
"n.load_file(\"import3d.hoc\")\n",
"\n",
"## load the morphology and instantiate at the top level (i.e. not in a class)\n",
"cell = n.Import3d_SWC_read()\n",
"cell.input(\"c91662.CNG.swc\")\n",
"n.Import3d_GUI(cell, 0)\n",
"i3d = n.Import3d_GUI(cell, 0)\n",
"i3d.instantiate(None) # pass in a class to instantiate inside the class instead\n",
"\n",
"## increase the number of segments\n",
"for sec in n.allsec():\n",
" sec.nseg = 1 + 2 * int(sec.L / 20)\n",
"\n",
"soma = n.soma[0]\n",
"\n",
"\n",
"def my_initial(node):\n",
" # return a certain function of the distance\n",
" return 2 * n.tanh(n.distance(soma(0.5), node) / 1000.0)\n",
"\n",
"\n",
"cyt = rxd.Region(n.allsec(), name=\"cyt\", nrn_region=\"i\")\n",
"ca = rxd.Species(cyt, name=\"ca\", charge=2, initial=my_initial)\n",
"\n",
"n.finitialize(-65 * mV)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"execution": {
"iopub.execute_input": "2025-08-18T03:36:18.730210Z",
"iopub.status.busy": "2025-08-18T03:36:18.729568Z",
"iopub.status.idle": "2025-08-18T03:36:33.952622Z",
"shell.execute_reply": "2025-08-18T03:36:33.952134Z"
}
},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "e49741c399fa43dfbdbd34a934aa60ca",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"FigureWidgetWithNEURON({\n",
" 'data': [{'hovertemplate': 'soma[0](0.5)
0.000',\n",
" 'line': {'color': '#00ffff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '22a73b54-f7df-45ea-895a-5321fb9ff95d',\n",
" 'x': [-8.86769962310791, 0.0, 8.86769962310791],\n",
" 'y': [0.0, 0.0, 0.0],\n",
" 'z': [0.0, 0.0, 0.0]},\n",
" {'hovertemplate': 'axon[0](0.00909091)
0.010',\n",
" 'line': {'color': '#01feff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '0f0e2044-1455-4de2-b96a-c1db3650b449',\n",
" 'x': [-5.329999923706055, -6.094121333400853],\n",
" 'y': [-5.349999904632568, -11.292340735151637],\n",
" 'z': [-3.630000114440918, -11.805355299531167]},\n",
" {'hovertemplate': 'axon[0](0.0272727)
0.030',\n",
" 'line': {'color': '#03fcff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '59229560-f214-469c-8625-98c52d39658a',\n",
" 'x': [-6.094121333400853, -6.360000133514404, -6.75,\n",
" -7.1118314714518265],\n",
" 'y': [-11.292340735151637, -13.359999656677246, -16.75,\n",
" -17.085087246355876],\n",
" 'z': [-11.805355299531167, -14.649999618530273, -19.270000457763672,\n",
" -19.981077971228146]},\n",
" {'hovertemplate': 'axon[0](0.0454545)
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" 'line': {'color': '#06f9ff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '156fcad9-b5ea-4185-83cb-34f31023d7f5',\n",
" 'x': [-7.1118314714518265, -9.050000190734863, -7.8939691125139095],\n",
" 'y': [-17.085087246355876, -18.8799991607666, -20.224003038013603],\n",
" 'z': [-19.981077971228146, -23.790000915527344, -28.996839067930022]},\n",
" {'hovertemplate': 'axon[0](0.0636364)
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" 'line': {'color': '#09f6ff', 'width': 2},\n",
" 'mode': 'lines',\n",
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" 'type': 'scatter3d',\n",
" 'uid': '2bc9d184-d7bf-48e3-948d-9104983574ad',\n",
" 'x': [-7.8939691125139095, -7.820000171661377, -6.989999771118164,\n",
" -9.244513598630135],\n",
" 'y': [-20.224003038013603, -20.309999465942383, -22.81999969482422,\n",
" -24.35991271814723],\n",
" 'z': [-28.996839067930022, -29.329999923706055, -34.04999923706055,\n",
" -37.46699683937078]},\n",
" {'hovertemplate': 'axon[0](0.0818182)
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" 'line': {'color': '#0bf4ff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '01c10118-fca3-4108-9457-3e50bfb062a2',\n",
" 'x': [-9.244513598630135, -11.470000267028809, -12.92143809645921],\n",
" 'y': [-24.35991271814723, -25.8799991607666, -28.950349592719142],\n",
" 'z': [-37.46699683937078, -40.84000015258789, -45.56415165743572]},\n",
" {'hovertemplate': 'axon[0](0.1)
0.111',\n",
" 'line': {'color': '#0ef1ff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '8de54a9f-9ac3-4ed3-8c1e-48ea4c885115',\n",
" 'x': [-12.92143809645921, -13.550000190734863, -17.227732134298183],\n",
" 'y': [-28.950349592719142, -30.280000686645508, -34.7463434475075],\n",
" 'z': [-45.56415165743572, -47.61000061035156, -52.562778077371306]},\n",
" {'hovertemplate': 'axon[0](0.118182)
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" 'line': {'color': '#10efff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': 'e8c125b3-84b5-439e-b9d1-d9d1dec631d6',\n",
" 'x': [-17.227732134298183, -18.540000915527344, -21.60001495171383],\n",
" 'y': [-34.7463434475075, -36.34000015258789, -40.43413031311105],\n",
" 'z': [-52.562778077371306, -54.33000183105469, -59.70619269769682]},\n",
" {'hovertemplate': 'axon[0](0.136364)
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" 'line': {'color': '#13ecff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': 'abca82e4-3da1-487f-bdc4-f1f2022286c2',\n",
" 'x': [-21.60001495171383, -23.600000381469727, -24.502645712175635],\n",
" 'y': [-40.43413031311105, -43.11000061035156, -45.643855890157845],\n",
" 'z': [-59.70619269769682, -63.220001220703125, -67.77191234032888]},\n",
" {'hovertemplate': 'axon[0](0.154545)
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" 'line': {'color': '#15eaff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '3eda685a-74f8-4edd-bcb6-d2feae15b4f8',\n",
" 'x': [-24.502645712175635, -25.0, -29.091788222104714],\n",
" 'y': [-45.643855890157845, -47.040000915527344, -50.7483704262943],\n",
" 'z': [-67.77191234032888, -70.27999877929688, -74.93493469773061]},\n",
" {'hovertemplate': 'axon[0](0.172727)
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" 'line': {'color': '#18e7ff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '7782d874-51a5-4e86-836c-c729009809f0',\n",
" 'x': [-29.091788222104714, -31.829999923706055, -33.209827051757856],\n",
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" 'z': [-74.93493469773061, -78.05000305175781, -81.71464479572988]},\n",
" {'hovertemplate': 'axon[0](0.190909)
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" 'line': {'color': '#1be4ff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
" 'uid': '4cca3d65-fa3f-4ad3-b40c-82572fc0a20f',\n",
" 'x': [-33.209827051757856, -34.29999923706055, -36.33646383783327],\n",
" 'y': [-56.805261963255965, -59.630001068115234, -64.47716110192778],\n",
" 'z': [-81.71464479572988, -84.61000061035156, -87.387849984506]},\n",
" {'hovertemplate': 'axon[0](0.209091)
0.232',\n",
" 'line': {'color': '#1de2ff', 'width': 2},\n",
" 'mode': 'lines',\n",
" 'name': '',\n",
" 'type': 'scatter3d',\n",
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"})"
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},
"execution_count": 10,
"metadata": {},
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],
"source": [
"import plotly\n",
"\n",
"ps = n.PlotShape(False)\n",
"ps.variable(ca[cyt])\n",
"ps.scale(0, 2)\n",
"ps.plot(plotly).show(renderer=\"notebook_connected\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Using position:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We continue the above example adding a new species, that is initialized based on the x-coordinate. This could happen, for example, on a platform with a nutrient or temperature gradient:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"execution": {
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"iopub.status.busy": "2025-08-18T03:36:33.957033Z",
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"shell.execute_reply": "2025-08-18T03:36:35.672512Z"
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},
"outputs": [
{
"data": {
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"1.0"
]
},
"execution_count": 11,
"metadata": {},
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],
"source": [
"def my_initial2(node):\n",
" # return a certain function of the x-coordinate\n",
" return 1 + n.tanh(node.x3d / 100.0)\n",
"\n",
"\n",
"alpha = rxd.Parameter(cyt, name=\"alpha\", initial=my_initial2)\n",
"\n",
"n.finitialize(-65 * mV)"
]
},
{
"cell_type": "code",
"execution_count": 12,
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"shell.execute_reply": "2025-08-18T03:36:50.454413Z"
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"import plotly\n",
"\n",
"ps = n.PlotShape(False)\n",
"ps.variable(alpha[cyt])\n",
"ps.scale(0, 2)\n",
"ps.plot(plotly).show(renderer=\"notebook_connected\")"
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"## Option 4: to steady state"
]
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"source": [
"Sometimes one might want to initialize a simulation to steady-state where e.g. diffusion, ion channel currents, and chemical reactions all balance each other out. There may be no such possible initial condition due to the interacting parts.\n",
"\n",
"In principle, such initial conditions could be assigned using a variant of the option 3 approach above. In practice, however, it may be simpler to omit the initial= keyword argument, and use an n.FInitializeHandler to loop over locations, setting the values for all states at a given location at the same time. A full example is beyond the scope of this tutorial."
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Mr=J.getTickFormat(mt),Fr=mt.ticklabelmode===\"period\";Fr&&(mt._rawTick0=mt.tick0),/%[uVW]/.test(Mr)?mt.tick0=e.dateTick0(mt.calendar,2):mt.tick0=e.dateTick0(mt.calendar,1),Fr&&(mt._dowTick0=mt.tick0)}}else Tr>u?mt.dtick=ur(gt,u,nt):Tr>y?mt.dtick=ur(gt,y,Qe):Tr>f?mt.dtick=ur(gt,f,Qe):(kr=br(10),mt.dtick=ur(gt,kr,tt))}else if(mt.type===\"log\"){mt.tick0=0;var Lr=e.simpleMap(mt.range,mt.r2l);if(mt._isMinor&&(gt*=1.5),gt>.7)mt.dtick=Math.ceil(gt);else if(Math.abs(Lr[1]-Lr[0])<1){var Jr=1.5*Math.abs((Lr[1]-Lr[0])/gt);gt=Math.abs(Math.pow(10,Lr[1])-Math.pow(10,Lr[0]))/Jr,kr=br(10),mt.dtick=\"L\"+ur(gt,kr,tt)}else mt.dtick=gt>.3?\"D2\":\"D1\"}else mt.type===\"category\"||mt.type===\"multicategory\"?(mt.tick0=0,mt.dtick=Math.ceil(Math.max(gt,1))):pa(mt)?(mt.tick0=0,kr=1,mt.dtick=ur(gt,kr,jt)):(mt.tick0=0,kr=br(10),mt.dtick=ur(gt,kr,tt));if(mt.dtick===0&&(mt.dtick=1),!x(mt.dtick)&&typeof mt.dtick!=\"string\"){var oa=mt.dtick;throw mt.dtick=1,\"ax.dtick error: \"+String(oa)}};function ar(mt){var gt=mt.dtick;if(mt._tickexponent=0,!x(gt)&&typeof gt!=\"string\"&&(gt=1),(mt.type===\"category\"||mt.type===\"multicategory\")&&(mt._tickround=null),mt.type===\"date\"){var Er=mt.r2l(mt.tick0),kr=mt.l2r(Er).replace(/(^-|i)/g,\"\"),br=kr.length;if(String(gt).charAt(0)===\"M\")br>10||kr.substr(5)!==\"01-01\"?mt._tickround=\"d\":mt._tickround=+gt.substr(1)%12===0?\"y\":\"m\";else if(gt>=b&&br<=10||gt>=b*15)mt._tickround=\"d\";else if(gt>=y&&br<=16||gt>=u)mt._tickround=\"M\";else if(gt>=f&&br<=19||gt>=y)mt._tickround=\"S\";else{var Tr=mt.l2r(Er+gt).replace(/^-/,\"\").length;mt._tickround=Math.max(br,Tr)-20,mt._tickround<0&&(mt._tickround=4)}}else if(x(gt)||gt.charAt(0)===\"L\"){var Mr=mt.range.map(mt.r2d||Number);x(gt)||(gt=Number(gt.substr(1))),mt._tickround=2-Math.floor(Math.log(gt)/Math.LN10+.01);var Fr=Math.max(Math.abs(Mr[0]),Math.abs(Mr[1])),Lr=Math.floor(Math.log(Fr)/Math.LN10+.01),Jr=mt.minexponent===void 0?3:mt.minexponent;Math.abs(Lr)>Jr&&(ke(mt.exponentformat)&&!Te(Lr)?mt._tickexponent=3*Math.round((Lr-1)/3):mt._tickexponent=Lr)}else mt._tickround=null}J.tickIncrement=function(mt,gt,Er,kr){var br=Er?-1:1;if(x(gt))return e.increment(mt,br*gt);var Tr=gt.charAt(0),Mr=br*Number(gt.substr(1));if(Tr===\"M\")return e.incrementMonth(mt,Mr,kr);if(Tr===\"L\")return Math.log(Math.pow(10,mt)+Mr)/Math.LN10;if(Tr===\"D\"){var Fr=gt===\"D2\"?Ot:St,Lr=mt+br*.01,Jr=e.roundUp(e.mod(Lr,1),Fr,Er);return Math.floor(Lr)+Math.log(g.round(Math.pow(10,Jr),1))/Math.LN10}throw\"unrecognized dtick \"+String(gt)},J.tickFirst=function(mt,gt){var Er=mt.r2l||Number,kr=e.simpleMap(mt.range,Er,void 0,void 0,gt),br=kr[1]=0&&Ba<=mt._length?ma:null};if(Tr&&e.isArrayOrTypedArray(mt.ticktext)){var ca=e.simpleMap(mt.range,mt.r2l),kt=(Math.abs(ca[1]-ca[0])-(mt._lBreaks||0))/1e4;for(Jr=0;Jr\"+Fr;else{var Jr=Ea(mt),oa=mt._trueSide||mt.side;(!Jr&&oa===\"top\"||Jr&&oa===\"bottom\")&&(Mr+=\"
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kt=String(gt.text).charAt(0);(kt===\"0\"||kt===\"1\")&&(mt._id.charAt(0)===\"y\"?gt.dx-=gt.fontSize/4:(gt.dy+=gt.fontSize/2,gt.dx+=(mt.range[1]>mt.range[0]?1:-1)*gt.fontSize*(Mr<0?.5:.25)))}}function yt(mt,gt){var Er=mt._categories[Math.round(gt.x)];Er===void 0&&(Er=\"\"),gt.text=String(Er)}function Fe(mt,gt,Er){var kr=Math.round(gt.x),br=mt._categories[kr]||[],Tr=br[1]===void 0?\"\":String(br[1]),Mr=br[0]===void 0?\"\":String(br[0]);Er?gt.text=Mr+\" - \"+Tr:(gt.text=Tr,gt.text2=Mr)}function Ke(mt,gt,Er,kr,br){br===\"never\"?br=\"\":mt.showexponent===\"all\"&&Math.abs(gt.x/mt.dtick)<1e-6&&(br=\"hide\"),gt.text=Le(gt.x,mt,br,kr)}function Ne(mt,gt,Er,kr,br){if(mt.thetaunit===\"radians\"&&!Er){var Tr=gt.x/180;if(Tr===0)gt.text=\"0\";else{var Mr=Ee(Tr);if(Mr[1]>=100)gt.text=Le(e.deg2rad(gt.x),mt,br,kr);else{var Fr=gt.x<0;Mr[1]===1?Mr[0]===1?gt.text=\"\\u03C0\":gt.text=Mr[0]+\"\\u03C0\":gt.text=[\"\",Mr[0],\"\",\"\\u2044\",\"\",Mr[1],\"\",\"\\u03C0\"].join(\"\"),Fr&&(gt.text=z+gt.text)}}}else gt.text=Le(gt.x,mt,br,kr)}function Ee(mt){function gt(Fr,Lr){return Math.abs(Fr-Lr)<=1e-6}function Er(Fr,Lr){return gt(Lr,0)?Fr:Er(Lr,Fr%Lr)}function kr(Fr){for(var Lr=1;!gt(Math.round(Fr*Lr)/Lr,Fr);)Lr*=10;return Lr}var br=kr(mt),Tr=mt*br,Mr=Math.abs(Er(Tr,br));return[Math.round(Tr/Mr),Math.round(br/Mr)]}var Ve=[\"f\",\"p\",\"n\",\"\\u03BC\",\"m\",\"\",\"k\",\"M\",\"G\",\"T\"];function ke(mt){return mt===\"SI\"||mt===\"B\"}function Te(mt){return mt>14||mt<-15}function Le(mt,gt,Er,kr){var br=mt<0,Tr=gt._tickround,Mr=Er||gt.exponentformat||\"B\",Fr=gt._tickexponent,Lr=J.getTickFormat(gt),Jr=gt.separatethousands;if(kr){var oa={exponentformat:Mr,minexponent:gt.minexponent,dtick:gt.showexponent===\"none\"?gt.dtick:x(mt)&&Math.abs(mt)||1,range:gt.showexponent===\"none\"?gt.range.map(gt.r2d):[0,mt||1]};ar(oa),Tr=(Number(oa._tickround)||0)+4,Fr=oa._tickexponent,gt.hoverformat&&(Lr=gt.hoverformat)}if(Lr)return gt._numFormat(Lr)(mt).replace(/-/g,z);var ca=Math.pow(10,-Tr)/2;if(Mr===\"none\"&&(Fr=0),mt=Math.abs(mt),mt\"+mr+\"\":Mr===\"B\"&&Fr===9?mt+=\"B\":ke(Mr)&&(mt+=Ve[Fr/3+5])}return br?z+mt:mt}J.getTickFormat=function(mt){var gt;function Er(Lr){return typeof Lr!=\"string\"?Lr:Number(Lr.replace(\"M\",\"\"))*S}function kr(Lr,Jr){var oa=[\"L\",\"D\"];if(typeof Lr==typeof Jr){if(typeof Lr==\"number\")return Lr-Jr;var ca=oa.indexOf(Lr.charAt(0)),kt=oa.indexOf(Jr.charAt(0));return ca===kt?Number(Lr.replace(/(L|D)/g,\"\"))-Number(Jr.replace(/(L|D)/g,\"\")):ca-kt}else return typeof Lr==\"number\"?1:-1}function br(Lr,Jr,oa){var ca=oa||function(mr){return mr},kt=Jr[0],ir=Jr[1];return(!kt&&typeof kt!=\"number\"||ca(kt)<=ca(Lr))&&(!ir&&typeof ir!=\"number\"||ca(ir)>=ca(Lr))}function Tr(Lr,Jr){var oa=Jr[0]===null,ca=Jr[1]===null,kt=kr(Lr,Jr[0])>=0,ir=kr(Lr,Jr[1])<=0;return(oa||kt)&&(ca||ir)}var Mr,Fr;if(mt.tickformatstops&&mt.tickformatstops.length>0)switch(mt.type){case\"date\":case\"linear\":{for(gt=0;gt=0&&br.unshift(br.splice(oa,1).shift())}});var Fr={false:{left:0,right:0}};return e.syncOrAsync(br.map(function(Lr){return function(){if(Lr){var Jr=J.getFromId(mt,Lr);Er||(Er={}),Er.axShifts=Fr,Er.overlayingShiftedAx=Mr;var oa=J.drawOne(mt,Jr,Er);return Jr._shiftPusher&&qa(Jr,Jr._fullDepth||0,Fr,!0),Jr._r=Jr.range.slice(),Jr._rl=e.simpleMap(Jr._r,Jr.r2l),oa}}}))},J.drawOne=function(mt,gt,Er){Er=Er||{};var kr=Er.axShifts||{},br=Er.overlayingShiftedAx||[],Tr,Mr,Fr;gt.setScale();var Lr=mt._fullLayout,Jr=gt._id,oa=Jr.charAt(0),ca=J.counterLetter(Jr),kt=Lr._plots[gt._mainSubplot];if(!kt)return;if(gt._shiftPusher=gt.autoshift||br.indexOf(gt._id)!==-1||br.indexOf(gt.overlaying)!==-1,gt._shiftPusher>.anchor===\"free\"){var ir=gt.linewidth/2||0;gt.ticks===\"inside\"&&(ir+=gt.ticklen),qa(gt,ir,kr,!0),qa(gt,gt.shift||0,kr,!1)}(Er.skipTitle!==!0||gt._shift===void 0)&&(gt._shift=ya(gt,kr));var mr=kt[oa+\"axislayer\"],$r=gt._mainLinePosition,ma=$r+=gt._shift,Ba=gt._mainMirrorPosition,Ca=gt._vals=J.calcTicks(gt),da=[gt.mirror,ma,Ba].join(\"_\");for(Tr=0;Tr0?Xo.bottom-Kn:0,gs))));var yl=0,Bu=0;if(gt._shiftPusher&&(yl=Math.max(gs,Xo.height>0?ji===\"l\"?Kn-Xo.left:Xo.right-Kn:0),gt.title.text!==Lr._dfltTitle[oa]&&(Bu=(gt._titleStandoff||0)+(gt._titleScoot||0),ji===\"l\"&&(Bu+=Aa(gt))),gt._fullDepth=Math.max(yl,Bu)),gt.automargin){Un={x:0,y:0,r:0,l:0,t:0,b:0};var El=[0,1],Vs=typeof gt._shift==\"number\"?gt._shift:0;if(oa===\"x\"){if(ji===\"b\"?Un[ji]=gt._depth:(Un[ji]=gt._depth=Math.max(Xo.width>0?Kn-Xo.top:0,gs),El.reverse()),Xo.width>0){var Jl=Xo.right-(gt._offset+gt._length);Jl>0&&(Un.xr=1,Un.r=Jl);var Nu=gt._offset-Xo.left;Nu>0&&(Un.xl=0,Un.l=Nu)}}else if(ji===\"l\"?(gt._depth=Math.max(Xo.height>0?Kn-Xo.left:0,gs),Un[ji]=gt._depth-Vs):(gt._depth=Math.max(Xo.height>0?Xo.right-Kn:0,gs),Un[ji]=gt._depth+Vs,El.reverse()),Xo.height>0){var Ic=Xo.bottom-(gt._offset+gt._length);Ic>0&&(Un.yb=0,Un.b=Ic);var Xu=gt._offset-Xo.top;Xu>0&&(Un.yt=1,Un.t=Xu)}Un[ca]=gt.anchor===\"free\"?gt.position:gt._anchorAxis.domain[El[0]],gt.title.text!==Lr._dfltTitle[oa]&&(Un[ji]+=Aa(gt)+(gt.title.standoff||0)),gt.mirror&>.anchor!==\"free\"&&(Wl={x:0,y:0,r:0,l:0,t:0,b:0},Wl[To]=gt.linewidth,gt.mirror&>.mirror!==!0&&(Wl[To]+=gs),gt.mirror===!0||gt.mirror===\"ticks\"?Wl[ca]=gt._anchorAxis.domain[El[1]]:(gt.mirror===\"all\"||gt.mirror===\"allticks\")&&(Wl[ca]=[gt._counterDomainMin,gt._counterDomainMax][El[1]]))}ml&&(Zu=M.getComponentMethod(\"rangeslider\",\"autoMarginOpts\")(mt,gt)),typeof gt.automargin==\"string\"&&(rt(Un,gt.automargin),rt(Wl,gt.automargin)),A.autoMargin(mt,ni(gt),Un),A.autoMargin(mt,Wt(gt),Wl),A.autoMargin(mt,zt(gt),Zu)}),e.syncOrAsync(cs)}};function rt(mt,gt){if(mt){var Er=Object.keys(W).reduce(function(kr,br){return gt.indexOf(br)!==-1&&W[br].forEach(function(Tr){kr[Tr]=1}),kr},{});Object.keys(mt).forEach(function(kr){Er[kr]||(kr.length===1?mt[kr]=0:delete mt[kr])})}}function dt(mt,gt){var Er=[],kr,br=function(Tr,Mr){var Fr=Tr.xbnd[Mr];Fr!==null&&Er.push(e.extendFlat({},Tr,{x:Fr}))};if(gt.length){for(kr=0;krmt.range[1],Fr=mt.ticklabelposition&&mt.ticklabelposition.indexOf(\"inside\")!==-1,Lr=!Fr;if(Er){var Jr=Mr?-1:1;Er=Er*Jr}if(kr){var oa=mt.side,ca=Fr&&(oa===\"top\"||oa===\"left\")||Lr&&(oa===\"bottom\"||oa===\"right\")?1:-1;kr=kr*ca}return mt._id.charAt(0)===\"x\"?function(kt){return t(br+mt._offset+mt.l2p(Gt(kt))+Er,Tr+kr)}:function(kt){return t(Tr+kr,br+mt._offset+mt.l2p(Gt(kt))+Er)}};function Gt(mt){return mt.periodX!==void 0?mt.periodX:mt.x}function Kt(mt){var gt=mt.ticklabelposition||\"\",Er=function(ir){return gt.indexOf(ir)!==-1},kr=Er(\"top\"),br=Er(\"left\"),Tr=Er(\"right\"),Mr=Er(\"bottom\"),Fr=Er(\"inside\"),Lr=Mr||br||kr||Tr;if(!Lr&&!Fr)return[0,0];var Jr=mt.side,oa=Lr?(mt.tickwidth||0)/2:0,ca=$,kt=mt.tickfont?mt.tickfont.size:12;return(Mr||kr)&&(oa+=kt*se,ca+=(mt.linewidth||0)/2),(br||Tr)&&(oa+=(mt.linewidth||0)/2,ca+=$),Fr&&Jr===\"top\"&&(ca-=kt*(1-se)),(br||kr)&&(oa=-oa),(Jr===\"bottom\"||Jr===\"right\")&&(ca=-ca),[Lr?oa:0,Fr?ca:0]}J.makeTickPath=function(mt,gt,Er,kr){kr||(kr={});var br=kr.minor;if(br&&!mt.minor)return\"\";var Tr=kr.len!==void 0?kr.len:br?mt.minor.ticklen:mt.ticklen,Mr=mt._id.charAt(0),Fr=(mt.linewidth||1)/2;return Mr===\"x\"?\"M0,\"+(gt+Fr*Er)+\"v\"+Tr*Er:\"M\"+(gt+Fr*Er)+\",0h\"+Tr*Er},J.makeLabelFns=function(mt,gt,Er){var kr=mt.ticklabelposition||\"\",br=function(Cn){return kr.indexOf(Cn)!==-1},Tr=br(\"top\"),Mr=br(\"left\"),Fr=br(\"right\"),Lr=br(\"bottom\"),Jr=Lr||Mr||Tr||Fr,oa=br(\"inside\"),ca=kr===\"inside\"&&mt.ticks===\"inside\"||!oa&&mt.ticks===\"outside\"&&mt.tickson!==\"boundaries\",kt=0,ir=0,mr=ca?mt.ticklen:0;if(oa?mr*=-1:Jr&&(mr=0),ca&&(kt+=mr,Er)){var $r=e.deg2rad(Er);kt=mr*Math.cos($r)+1,ir=mr*Math.sin($r)}mt.showticklabels&&(ca||mt.showline)&&(kt+=.2*mt.tickfont.size),kt+=(mt.linewidth||1)/2*(oa?-1:1);var ma={labelStandoff:kt,labelShift:ir},Ba,Ca,da,Sa,Ti=0,ai=mt.side,an=mt._id.charAt(0),sn=mt.tickangle,Mn;if(an===\"x\")Mn=!oa&&ai===\"bottom\"||oa&&ai===\"top\",Sa=Mn?1:-1,oa&&(Sa*=-1),Ba=ir*Sa,Ca=gt+kt*Sa,da=Mn?1:-.2,Math.abs(sn)===90&&(oa?da+=ue:sn===-90&&ai===\"bottom\"?da=se:sn===90&&ai===\"top\"?da=ue:da=.5,Ti=ue/2*(sn/90)),ma.xFn=function(Cn){return Cn.dx+Ba+Ti*Cn.fontSize},ma.yFn=function(Cn){return Cn.dy+Ca+Cn.fontSize*da},ma.anchorFn=function(Cn,Lo){if(Jr){if(Mr)return\"end\";if(Fr)return\"start\"}return!x(Lo)||Lo===0||Lo===180?\"middle\":Lo*Sa<0!==oa?\"end\":\"start\"},ma.heightFn=function(Cn,Lo,Xi){return Lo<-60||Lo>60?-.5*Xi:mt.side===\"top\"!==oa?-Xi:0};else if(an===\"y\"){if(Mn=!oa&&ai===\"left\"||oa&&ai===\"right\",Sa=Mn?1:-1,oa&&(Sa*=-1),Ba=kt,Ca=ir*Sa,da=0,!oa&&Math.abs(sn)===90&&(sn===-90&&ai===\"left\"||sn===90&&ai===\"right\"?da=se:da=.5),oa){var Bn=x(sn)?+sn:0;if(Bn!==0){var Qn=e.deg2rad(Bn);Ti=Math.abs(Math.sin(Qn))*se*Sa,da=0}}ma.xFn=function(Cn){return Cn.dx+gt-(Ba+Cn.fontSize*da)*Sa+Ti*Cn.fontSize},ma.yFn=function(Cn){return Cn.dy+Ca+Cn.fontSize*ue},ma.anchorFn=function(Cn,Lo){return x(Lo)&&Math.abs(Lo)===90?\"middle\":Mn?\"end\":\"start\"},ma.heightFn=function(Cn,Lo,Xi){return mt.side===\"right\"&&(Lo*=-1),Lo<-30?-Xi:Lo<30?-.5*Xi:0}}return ma};function sr(mt){return[mt.text,mt.x,mt.axInfo,mt.font,mt.fontSize,mt.fontColor].join(\"_\")}J.drawTicks=function(mt,gt,Er){Er=Er||{};var kr=gt._id+\"tick\",br=[].concat(gt.minor&>.minor.ticks?Er.vals.filter(function(Mr){return 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