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SYDE 556 Lecture 5 Dynamics.ipynb
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SYDE 556 Lecture 5 Dynamics.ipynb
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{
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{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# SYDE 556/750: Simulating Neurobiological Systems\n",
"\n",
"Accompanying Readings: Chapter 8\n",
"\n",
"ADMIN STUFF: Next assignment, project proposal due dates"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"\n",
"## Dynamics\n",
"\n",
"- Everything we've looked at so far has been feedforward\n",
" - There's some pattern of activity in one group of neurons representing $x$\n",
" - We want that to cause some pattern of activity in another group of neurons to represent $y=f(x)$\n",
" - These can be chained together to make more complex systems $z=h(f(x)+g(y))$\n",
"- What about recurrent networks?\n",
" - What happens when we connect a neural group back to itself?\n",
" \n",
"<img src=\"files/lecture5/recnet1.png\">\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Recurrent functions\n",
"\n",
"- What if we do exactly what we've done so far in the past, but instead of connecting one group of neurons to another, we just connect it back to itself\n",
" - Instead of $y=f(x)$\n",
" - We get $x=f(x)$ (???)\n",
"- As written, this is clearly non-sensical\n",
" - For example, if we do $f(x)=x+1$ then we'd have $x=x+1$, or $x-x=1$, or $0=1$\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- But don't forget about time\n",
" - What if it was $x_{t+1} = f(x_t)$\n",
" - Which makes more sense because we're talking about a real physical system\n",
" - This is a lot like a differential equation\n",
" - What would happen if we built this?"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Try it out\n",
"\n",
"- Let's try implementing this kind of circuit\n",
"- Start with $x_{t+1}=x_t+1$ "
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Populating the interactive namespace from numpy and matplotlib\n"
]
}
],
"source": [
"%pylab inline"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
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\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x113b9feb8>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"import nengo\n",
"\n",
"model = nengo.Network()\n",
"\n",
"with model:\n",
" ensA = nengo.Ensemble(100, dimensions=1)\n",
" \n",
" def feedback(x):\n",
" return x+1\n",
" \n",
" conn = nengo.Connection(ensA, ensA, function=feedback, synapse = 0.1)\n",
"\n",
" ensA_p = nengo.Probe(ensA, synapse=.01)\n",
" \n",
"sim = nengo.Simulator(model)\n",
"sim.run(.5)\n",
"\n",
"plot(sim.trange(), sim.data[ensA_p])\n",
"ylim(-1.5,1.5);"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- That sort of makes sense\n",
" - $x$ increases quickly, then hits an upper bound\n",
"- How quickly?\n",
" - What parameters of the system affect this?\n",
"- What are the precise dynamics?\n",
"\n",
"- What about $f(x)=-x$?"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
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],
"text/plain": [
"HtmlProgressBar cannot be displayed. Please use the TerminalProgressBar. It can be enabled with `nengo.rc.set('progress', 'progress_bar', 'nengo.utils.progress.TerminalProgressBar')`."
]
},
"metadata": {},
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"text/plain": [
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\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x113be37b8>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"with model:\n",
" def feedback(x):\n",
" return -x\n",
" \n",
" conn.function = feedback\n",
"\n",
"sim = nengo.Simulator(model)\n",
"sim.run(.5)\n",
"\n",
"plot(sim.trange(), sim.data[ensA_p])\n",
"ylim(-1.5,1.5);"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- That also makes sense. What if we nudge it away from zero?"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" <div id=\"44a0d57d-101b-43f5-84cd-a78b412135f1\" style=\"\n",
" width: 100%;\n",
" border: 1px solid #cfcfcf;\n",
" border-radius: 4px;\n",
" text-align: center;\n",
" position: relative;\">\n",
" <div class=\"pb-text\" style=\"\n",
" position: absolute;\n",
" width: 100%;\">\n",
" 0%\n",
" </div>\n",
" <div class=\"pb-fill\" style=\"\n",
" background-color: #bdd2e6;\n",
" width: 0%;\">\n",
" <style type=\"text/css\" scoped=\"scoped\">\n",
" @keyframes pb-fill-anim {\n",
" 0% { background-position: 0 0; }\n",
" 100% { background-position: 100px 0; }\n",
" }\n",
" </style>\n",
" \n",
" </div>\n",
" </div>"
],
"text/plain": [
"HtmlProgressBar cannot be displayed. Please use the TerminalProgressBar. It can be enabled with `nengo.rc.set('progress', 'progress_bar', 'nengo.utils.progress.TerminalProgressBar')`."
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/javascript": [
"\n",
" (function () {\n",
" var root = document.getElementById('44a0d57d-101b-43f5-84cd-a78b412135f1');\n",
" var text = root.getElementsByClassName('pb-text')[0];\n",
" var fill = root.getElementsByClassName('pb-fill')[0];\n",
"\n",
" text.innerHTML = 'Build finished in 0:00:01.';\n",
" \n",
" fill.style.width = '100%';\n",
" fill.style.animation = 'pb-fill-anim 2s linear infinite';\n",
" fill.style.backgroundSize = '100px 100%';\n",
" fill.style.backgroundImage = 'repeating-linear-gradient(' +\n",
" '90deg, #bdd2e6, #edf2f8 40%, #bdd2e6 80%, #bdd2e6)';\n",
" \n",
" \n",
" fill.style.animation = 'none';\n",
" fill.style.backgroundImage = 'none';\n",
" \n",
" })();\n",
" "
],
"text/plain": [
"<IPython.core.display.Javascript object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
"\n",
" <div id=\"332365f3-39b9-4568-a0f9-3fc07342a5f1\" style=\"\n",
" width: 100%;\n",
" border: 1px solid #cfcfcf;\n",
" border-radius: 4px;\n",
" text-align: center;\n",
" position: relative;\">\n",
" <div class=\"pb-text\" style=\"\n",
" position: absolute;\n",
" width: 100%;\">\n",
" 0%\n",
" </div>\n",
" <div class=\"pb-fill\" style=\"\n",
" background-color: #bdd2e6;\n",
" width: 0%;\">\n",
" <style type=\"text/css\" scoped=\"scoped\">\n",
" @keyframes pb-fill-anim {\n",
" 0% { background-position: 0 0; }\n",
" 100% { background-position: 100px 0; }\n",
" }\n",
" </style>\n",
" \n",
" </div>\n",
" </div>"
],
"text/plain": [
"HtmlProgressBar cannot be displayed. Please use the TerminalProgressBar. It can be enabled with `nengo.rc.set('progress', 'progress_bar', 'nengo.utils.progress.TerminalProgressBar')`."
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/javascript": [
"\n",
" (function () {\n",
" var root = document.getElementById('332365f3-39b9-4568-a0f9-3fc07342a5f1');\n",
" var text = root.getElementsByClassName('pb-text')[0];\n",
" var fill = root.getElementsByClassName('pb-fill')[0];\n",
"\n",
" text.innerHTML = 'Simulation finished in 0:00:01.';\n",
" \n",
" if (100.0 > 0.) {\n",
" fill.style.transition = 'width 0.1s linear';\n",
" } else {\n",
" fill.style.transition = 'none';\n",
" }\n",
"\n",
" fill.style.width = '100.0%';\n",
" fill.style.animation = 'none';\n",
" fill.style.backgroundImage = 'none'\n",
" \n",
" \n",
" fill.style.animation = 'none';\n",
" fill.style.backgroundImage = 'none';\n",
" \n",
" })();\n",
" "
],
"text/plain": [
"<IPython.core.display.Javascript object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x118a775f8>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"from nengo.utils.functions import piecewise\n",
"\n",
"with model:\n",
" stim = nengo.Node(piecewise({0:1, .2:-1, .4:0}))\n",
" nengo.Connection(stim, ensA)\n",
" \n",
"sim = nengo.Simulator(model)\n",
"sim.run(.6)\n",
"\n",
"plot(sim.trange(), sim.data[ensA_p])\n",
"ylim(-1.5,1.5);"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- With an input of 1, $x=0.5$\n",
"- With an input of -1, $x=-0.5$\n",
"- With an input of 0, it goes back to $x=0$\n",
"- Does this make sense?\n",
" - Why / why not?\n",
" - And why that particular timing/curvature?\n",
" \n",
"- What about $f(x)=x^2$? "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
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" </style>\n",
" \n",
" </div>\n",
" </div>"
],
"text/plain": [
"HtmlProgressBar cannot be displayed. Please use the TerminalProgressBar. It can be enabled with `nengo.rc.set('progress', 'progress_bar', 'nengo.utils.progress.TerminalProgressBar')`."
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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" fill.style.animation = 'none';\n",
" fill.style.backgroundImage = 'none';\n",
" \n",
" })();\n",
" "
],
"text/plain": [
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]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
"\n",
" <div id=\"ebecdc33-46b3-4218-a632-e740fc8ca88f\" style=\"\n",
" width: 100%;\n",
" border: 1px solid #cfcfcf;\n",
" border-radius: 4px;\n",
" text-align: center;\n",
" position: relative;\">\n",
" <div class=\"pb-text\" style=\"\n",
" position: absolute;\n",
" width: 100%;\">\n",
" 0%\n",
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" <div class=\"pb-fill\" style=\"\n",
" background-color: #bdd2e6;\n",
" width: 0%;\">\n",
" <style type=\"text/css\" scoped=\"scoped\">\n",
" @keyframes pb-fill-anim {\n",
" 0% { background-position: 0 0; }\n",
" 100% { background-position: 100px 0; }\n",
" }\n",
" </style>\n",
" \n",
" </div>\n",
" </div>"
],
"text/plain": [
"HtmlProgressBar cannot be displayed. Please use the TerminalProgressBar. It can be enabled with `nengo.rc.set('progress', 'progress_bar', 'nengo.utils.progress.TerminalProgressBar')`."
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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"\n",
" text.innerHTML = 'Simulation finished in 0:00:01.';\n",
" \n",
" if (100.0 > 0.) {\n",
" fill.style.transition = 'width 0.1s linear';\n",
" } else {\n",
" fill.style.transition = 'none';\n",
" }\n",
"\n",
" fill.style.width = '100.0%';\n",
" fill.style.animation = 'none';\n",
" fill.style.backgroundImage = 'none'\n",
" \n",
" \n",
" fill.style.animation = 'none';\n",
" fill.style.backgroundImage = 'none';\n",
" \n",
" })();\n",
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],
"text/plain": [
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]
},
"metadata": {},
"output_type": "display_data"
},
{
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\n",
"text/plain": [
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]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"with model:\n",
" stim.output = piecewise({.1:.2, .2:.4, .5:0})\n",
" def feedback(x):\n",
" return x*x\n",
" \n",
" conn.function = feedback\n",
"\n",
"sim = nengo.Simulator(model)\n",
"sim.run(.6)\n",
"\n",
"plot(sim.trange(), sim.data[ensA_p])\n",
"ylim(-1.5,1.5); "
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Well that's weird\n",
" - Stable at $x=0$ with no input \n",
" - Stable at .2 \n",
" - Unstable at .4, shoots up high\n",
" - Something very strange happens around $x=1$ when the input is turned off (why decay if $f(x) = x^2$?)\n",
"- Why is this happening?\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Making sense of dynamics\n",
"\n",
"- Let's go back to something simple\n",
"- Just a single feed-forward neural population\n",
" - Encode $x$ into current, compute spikes, decode filtered spikes into $\\hat{x}$\n",
"- Instead of a constant input, let's change the input\n",
" - Change it suddenly from zero to one to get a sense of what's happening with changes"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
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