![](\"files/lecture4/communication1.png\")
"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Connecting neurons\n",
"\n",
"- Up till now, we've always had the current going into a neuron be something we computed from $x$\n",
" - $J = \\alpha e \\cdot x + J^{bias}$\n",
"- This will continue to be how we handle inputs\n",
" - Sensory neurons, for example\n",
" - Or whatever's coming from the rest of the brain that we're not modelling (yet)\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- But what about other groups of neurons?\n",
" - How do they end up getting the amount of input current that we're injecting with $J = \\alpha e \\cdot x + J^{bias}$ ?\n",
" - Where does that current come from?\n",
" - Inputs from neurons connected to this one\n",
" - Through weighted synaptic connections\n",
" - Let's think about neurons in a simple case"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### A communication channel\n",
"\n",
"- Let's say we have two groups of neurons\n",
" - One group represents $x$\n",
" - One group represents $y$\n",
" - Can we pass the value from one group of neurons to the other? \n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Without worrying about biological plausibility to start, we can formulate this in two steps\n",
" - Drive the first population $a$ with the input, $x$, then decoded it to give $\\hat{x}$\n",
" - Now use $y=\\hat{x}$ to drive the 2nd population $b$, and then decode that\n",
"- Let's start by first constructing the two populations\n",
" - Stimulate them both directly and decode to compare\n",
" \n",
"![](\"files/lecture4/comm_channel.png\")
"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"%pylab inline"
]
},
{
"cell_type": "code",
"execution_count": 23,
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\n",
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" \n",
"
\n",
" \n", "`upsilon_f=np.dot(A.T, f_x)`
\n", "`d_f = np.dot(np.linalg.pinv(gamma),upsilon)`\n", "\n", "`f_xhat = np.dot(A, d_f)`" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "- We call standard $d_i$ \"representational decoders\"\n", "- We call $d_i^{f(x)}$ \"transformational decoders\" (or \"decoders for $f(x)$\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Adding\n", "\n", "- What if we want to combine the inputs from two different populations?\n", " - Linear case: $z=x+y$\n", "\n", "
![](\"files/lecture4/adding1.png\")
\n"
]
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{
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"source": [
"- We want the total current going into a $z$ neuron to be $J=\\alpha e \\cdot (x+y) + J^{bias}$\n",
"- How can we achieve this?\n",
"- Again, substitute into the equation, where $z = x+y \\approx \\hat{x}+\\hat{y}$\n",
" - $J_k=\\alpha_k e \\cdot (\\hat{x}+\\hat{y}) + J_k^{bias}$\n",
" - $\\hat{x} = \\sum_i a_i d_i$\n",
" - $\\hat{y} = \\sum_j a_j d_j$\n",
" - $J_k=\\alpha_k e_k \\cdot (\\sum_i a_i d_i+\\sum_j a_j d_j) + J_k^{bias}$\n",
" - $J_k=\\sum_i(\\alpha_k e_k \\cdot d_i a_i) + \\sum_j(\\alpha_k e_k \\cdot d_j a_j) + J_k^{bias}$\n",
" - $J_k=\\sum_i(\\omega_{ik} a_i) + \\sum_j(\\omega_{jk} a_j) + J_k^{bias}$\n",
" - $\\omega_{ik}=\\alpha_k e_k \\cdot d_i$ and $\\omega_{jk}=\\alpha_k e_k \\cdot d_j$ \n",
"- Putting multiple inputs into a neuron automatically gives us addition!"
]
},
{
"cell_type": "code",
"execution_count": 40,
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"source": [
"from nengo_gui.ipython import IPythonViz\n",
"IPythonViz(model)#, \"configs/mult.py.cfg\")"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- Multiplication is quite powerful, and has lots of uses\n",
" - Gating of signals\n",
" - Attention effects\n",
" - Binding\n",
" - Statistical inference\n",
" \n",
"- Here's a simple gating example using the same network"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"slideshow": {
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}
},
"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')`."
]
},
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{
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" (function () {\n",
" var root = document.getElementById('91462008-ba00-4244-a69b-cb9d172f4c6e');\n",
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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",
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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": {},
"output_type": "display_data"
},
{
"data": {
"application/javascript": [
"\n",
" (function () {\n",
" var root = document.getElementById('d160b43b-59e9-4a61-ac54-c3c422e25cd2');\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",
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" fill.style.transition = 'none';\n",
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"\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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"