![](\"files/lecture5/recnet1.png\")
\n",
"\n"
]
},
{
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"metadata": {
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"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": {
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}
},
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\n",
" ![](\"files/lecture5/control_sys.png\")
\n",
"- Our goal is to convert this to a structure which has $h(t)$ as the transfer function instead of the standard $\\int$\n",
"![](\"files/lecture5/control_sysh.png\")
\n",
"\n",
"\n"
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"source": [
"- Using Laplace on the standard form gives:\n",
"\n",
"$sX(s) = A X(s) + B U(s)$\n",
"\n",
"- Laplace on the 'neural control' form gives (as before where $F(s) = A'X(s) + B'U(s)$):\n",
"\n",
"$X(s) = {1 \\over {1 + s\\tau}} (A'X(s) + B'U(s))$\n",
"\n",
"$X(s) + \\tau sX(s) = (A'X(s) + B'U(s))$\n",
"\n",
"$sX(s) = {1 \\over \\tau} (A'X(s) + B'U(s) - X(s))$\n",
"\n",
"$sX(s) = {1 \\over \\tau} ((A' - I) X(s) + B'U(s))$\n",
"\n",
"- Making the 'standard' and 'neural' equations equal to one another, we find that for any system with a given A and B, the A' and B' of the equivalent neural system are given by:\n",
"\n",
"$A' = \\tau A + I$ and\n",
"\n",
"$B' = \\tau B$\n",
"\n",
"- where $I$ is the identity matrix\n",
"\n",
"- This is nice because lots of engineers think of the systems they build in these terms (i.e. as linear control systems)."
]
},
{
"cell_type": "markdown",
"metadata": {
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"slide_type": "subslide"
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},
"source": [
"\n",
"#### Nonlinear systems\n",
"- In fact, these same steps can be taken to account for nonlinear control systems as well:\n",
"\n",
"$\\dot{x}(t) = f(x(t),u(t),t)$\n",
"\n",
"- For a neural system with transfer function $h(t)$:\n",
"\n",
"$X(s) = H(s)F'(X(s),U(s),s)$\n",
"\n",
"$X(s) = {1 \\over {1 + s\\tau}} F'(X(s),U(s),s)$\n",
"\n",
"$sX(s) = {1 \\over \\tau} (F'(X(s),U(s),s) - X(s))$\n",
"\n",
"- This gives the general result (slightly more general than what we saw earlier):\n",
"\n",
"$F'(X(s),U(s),s) = \\tau(F(X(s),U(s),s)) + X(s)$"
]
},
{
"cell_type": "markdown",
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}
},
"source": [
"## Applications\n",
"\n",
"### Eye control\n",
"\n",
"- Part of the brainstem called the nuclei prepositus hypoglossi\n",
"- Input is eye velocity $v$\n",
"- Output is eye position $x$\n",
"- $\\dot{x}=v$\n",
" - This is an integrator ($x$ is the integral of $v$)\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- It's a linear system, so, to get it in the standard control form $\\dot{x}=Ax+Bu$ we have:\n",
" - $A=0$\n",
" - $B=1$\n",
"- So that means we need $A'=\\tau 0 + I = 1$ and $B'=\\tau 1 = \\tau$\n",
"![](\"files/lecture5/eye_sys.png\")
"
]
},
{
"cell_type": "code",
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\n",
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\n",
"\n",
"- What affects this?"
]
},
{
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"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Controlled Integrator\n",
"\n",
"- What if we want an integrator where we can adjust the decay on-the-fly?\n",
"- Separate input telling us what the decay constant $d$ should be\n",
"\n",
"$\\dot{x} = -d x + v$\n",
"\n",
"- So there are two inputs: $v$ and $d$\n",
"\n",
"\n",
"- This is no longer in the standard $Ax + Bu$ form. Sort of...\n",
" - Let $A = -d(t)$, so it's not a matrix\n",
" - But it is of the more general form: ${dx \\over dt}=f(x)+g(u)$\n",
" \n",
"- We need to compute a nonlinear function of an input ($d$) and the state variable ($x$)\n",
"- How can we do this?"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Going to 2D so we can compute the nonlinear function\n",
" - Let's have the state variable be $[x, d]$\n",
" \n",
"![](\"files/lecture5/controlled_integrator.png\")
"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"slideshow": {
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}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
"
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],
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"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"
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{
"data": {
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"\n",
" (function () {\n",
" var root = document.getElementById('0bd78810-6dd4-4dda-b1d6-04fe2ba2e423');\n",
" var text = root.getElementsByClassName('pb-text')[0];\n",
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"\n",
" text.innerHTML = 'Build finished in 0:00:01.';\n",
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" 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",
" "
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]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/javascript": [
"\n",
" (function () {\n",
" var root = document.getElementById('afdcf16c-8320-46dd-8e56-dd2a65abda3f');\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",
" "
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" "
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from nengo_gui.ipython import IPythonViz\n",
"IPythonViz(model, \"configs/controlled_integrator.py.cfg\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Other fun functions\n",
"\n",
"- Oscillator\n",
" - $F = -kx = m \\ddot{x}$ let $\\omega = \\sqrt{\\frac{k}{m}}$\n",
" - $\\frac{d}{dt} \\begin{bmatrix}\n",
"\\omega x \\\\\n",
"\\dot{x}\n",
"\\end{bmatrix}\n",
"=\n",
"\\begin{bmatrix}\n",
"0 & \\omega \\\\\n",
"-\\omega & 0\n",
"\\end{bmatrix}\n",
"\\begin{bmatrix}\n",
"x_0 \\\\\n",
"x_1\n",
"\\end{bmatrix}$\n",
" - Therefore, with the above, $\\dot{x}=[x_1, -x_0]$"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
"
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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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{
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" 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",
" "
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]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/javascript": [
"\n",
" (function () {\n",
" var root = document.getElementById('44099087-03f1-41ce-8611-51ed2023609f');\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",
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" } 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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" \n",
" \n",
"
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" "
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from nengo_gui.ipython import IPythonViz\n",
"IPythonViz(model, \"configs/oscillator.py.cfg\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Lorenz Attractor (a chaotic attractor)\n",
"\n",
" - $\\dot{x}=[10x_1-10x_0, -x_0 x_2-x_1, x_0 x_1 - {8 \\over 3}(x_2+28)-28]$\n"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
"
"
],
"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",
" (function () {\n",
" var root = document.getElementById('c496a023-ab9c-4e5d-89cf-eaf10ef6e80a');\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",
" "
],
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]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/javascript": [
"\n",
" (function () {\n",
" var root = document.getElementById('0dbbeada-fc65-436f-a95c-c43a407fa142');\n",
" var text = root.getElementsByClassName('pb-text')[0];\n",
" var fill = root.getElementsByClassName('pb-fill')[0];\n",
"\n",
" text.innerHTML = 'Simulation finished in 0:00:05.';\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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" \n",
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" "
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from nengo_gui.ipython import IPythonViz\n",
"IPythonViz(model, \"configs/lorenz.py.cfg\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Note: This is not the original Lorenz attractor. \n",
" - The original is $\\dot{x}=[10x_1-10x_0, x_0 (28-x_2)-x_1, x_0 x_1 - {8 \\over 3}(x_2)]$\n",
" - Why change it to $\\dot{x}=[10x_1-10x_0, -x_0 x_2-x_1, x_0 x_1 - {8 \\over 3}(x_2+28)-28]$?\n",
" - What's being changed here?"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Oscillators with different paths\n",
"\n",
"- Since we can implement any function, we're not limited to linear oscillators\n",
"- What about a \"square\" oscillator?\n",
" - Instead of the value going in a circle, it traces out a square\n",
"\n",
"$$\n",
"{\\dot{x}} = \\begin{cases}\n",
" [r, 0] &\\mbox{if } |x_1|>|x_0| \\wedge x_1>0 \\\\ \n",
" [-r, 0] &\\mbox{if } |x_1|>|x_0| \\wedge x_1<0 \\\\ \n",
" [0, -r] &\\mbox{if } |x_1|<|x_0| \\wedge x_0>0 \\\\ \n",
" [0, r] &\\mbox{if } |x_1|<|x_0| \\wedge x_0<0 \\\\ \n",
" \\end{cases}\n",
"$$ "
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
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"\n",
" \n",
"
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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')`."
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{
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" 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",
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"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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{
"data": {
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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from nengo_gui.ipython import IPythonViz\n",
"IPythonViz(model) #do config"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Does this do what you expect?\n",
"- How is it affected by:\n",
" - Number of neurons?\n",
" - Post-synaptic time constant?\n",
" - Decoding filter time constant?\n",
" - Speed of oscillation (``r``)?"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- What about this shape?\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
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],
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"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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{
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"\n",
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" 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",
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]
},
"metadata": {},
"output_type": "display_data"
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{
"data": {
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"\n",
" (function () {\n",
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" fill.style.animation = 'none';\n",
" fill.style.backgroundImage = 'none'\n",
" \n",
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},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from nengo_gui.ipython import IPythonViz\n",
"IPythonViz(model) #do config"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- We are doing things differently here\n",
"- The actual $x$ value is a normal circle oscillator\n",
"- The heart shape is a function of $x$\n",
" - But that's just a different decoder\n",
"- Would it be possible to do an oscillator where $x$ followed this shape?\n",
" - How could we tell them apart in terms of neural behaviour?\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Controlled Oscillator\n",
"\n",
"- Change the frequency of the oscillator on-the-fly\n",
"\n",
"- $\\dot{x}=[x_1 x_2, -x_0 x_2]$"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {
"slideshow": {
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}
},
"outputs": [
{
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" fill.style.animation = 'pb-fill-anim 2s linear infinite';\n",
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},
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{
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{
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"text": [
"ERROR:nengo_gui.server:Error response\n",
"Traceback (most recent call last):\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/server.py\", line 412, in do_GET\n",
" self.http_GET()\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/guibackend.py\", line 137, in http_GET\n",
" server.HttpWsRequestHandler.http_GET(self)\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/server.py\", line 428, in http_GET\n",
" response = getattr(self, command)()\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/guibackend.py\", line 86, in auth_checked\n",
" return fn(inst)\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/guibackend.py\", line 233, in serve_main\n",
" page = self.server.create_page(filename, reset_cfg=reset_cfg)\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/guibackend.py\", line 457, in create_page\n",
" reset_cfg=reset_cfg)\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/page.py\", line 114, in __init__\n",
" self.load()\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/page.py\", line 172, in load\n",
" self.name_finder = nengo_gui.NameFinder(self.locals, self.model)\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/namefinder.py\", line 13, in __init__\n",
" self.find_names(net)\n",
" File \"/Users/celiasmi/Documents/nengo/nengo_gui/nengo_gui/namefinder.py\", line 16, in find_names\n",
" net_name = self.known_name[net]\n",
"KeyError: