41:null 42:["$","title",null,{"children":"Adaptive LIF Neurons | Steve Oney | Observable"}] 48:T4a1,On top of this code, we will add: - `inh` to represent ${tex`w`}, a variable that tracks how much the threshold voltage (`v_th`) has increased (meaning that `v` will need to exceed `this.v_th + this.inh`) - `tau_inh` to represent ${tex`\tau_w`}, a time constant that tracks how `inh` should evolve over time - `inc_inh` to represent ${tex`b`}, which specifies how much to increase `inh` every time the neuron spikes We can update our constructor to include defaults for `tau_inh` and `inc_inh` (`0.05` and `1` respectively): ```javascript constructor(... tau_inh = 0.05, inc_inh = 1) ... ``` and initializing our instance variables (with `this.inh` initialized to `0`): ```javascript this.inh = 0; this.tau_inh = tau_inh; this.inc_inh = inc_inh; ``` Now, we want to update our firing condition (`if(this.v > this.v_th)`) to add `this.inh`: ```javascript if(this.v >= this.v_th + this.inh) ... ``` and we want to update `this.inh` every timestep: ```javascript this.inh = this.inh * (1 - t_step/this.tau_inh); ``` and if we fire: ```javascript if(this.v >= this.v_th + this.inh) { this.inh += this.inc_inh ... ``` so we end up with the following code:49:T4b2,class ALIF { constructor(tau_rc=0.2, v_init=0, tau_ref=0.002, v_th=1, tau_inh = 0.05, inc_inh = 1) { // <-- MODIFIED (added ", tau_inh = 0.05, inc_inh = 1") this.tau_rc = tau_rc; this.tau_ref = tau_ref; this.v = v_init; this.v_th = v_th; this.output = 0; this.refractory_time = 0; // vvv ADDED vvv this.inh = 0; this.tau_inh = tau_inh; this.inc_inh = inc_inh; // ^^^ ADDED ^^^ } step(I, t_step) { this.refractory_time -= t_step; // Subtract the amount of time thhat passed from our refractory time if(this.refractory_time < 0) { // If we aren't in our refractory period, update the voltage this.v = this.v * (1 - t_step/this.tau_rc) + (I * t_step / this.tau_rc); } // vvv ADDED vvv this.inh = this.inh * (1 - t_step/this.tau_inh); // ^^^ ADDED ^^^ if(this.v >= this.v_th + this.inh) { // <-- MODIFIED (added "+ this.inh") // vvv ADDED vvv this.inh += this.inc_inh; // ^^^ ADDED ^^^ this.refractory_time = this.tau_ref; // Set refractory time this.output = 1 / t_step; // Fire this.v = 0; // Reset potential } else { this.output = 0; // Don't fire } } }4a:T4e1,function plotSpikes(inp, tau_inh = 5, inc_inh = 0.2, runtime=0.8) { const TIME_STEP = 1e-3; const NUM_STEPS = runtime / TIME_STEP; // Simulate for runtime seconds const J = () => inp; const neuron = new ALIF(0.02, 0, 0.2); neuron.tau_inh = tau_inh; neuron.inc_inh = inc_inh; neuron.tau_ref = 1e-2; const v_history = [ {t: 0, v: neuron.v} ]; let numSpikes = 0; for(let i = 0; i < NUM_STEPS; i++) { const j = J((i+1)*TIME_STEP); neuron.step(j, TIME_STEP); const out = neuron.output * TIME_STEP; if(out > 0) { numSpikes++; } v_history.push({t: (i+1)*TIME_STEP, v: neuron.v, vth: neuron.v_th + neuron.inh, j, out }); } // const plot = Plot.line(v_history, {x: "t", y: "out"}).plot(); const plot = Plot.plot({ y: { grid: true, label: 'j' }, marks: [ Plot.line(v_history, {x: "t", y: "vth", stroke: "blue"}), Plot.line(v_history, {x: "t", y: "v", stroke: "red"}), Plot.line(v_history, {x: "t", y: "out", stroke: "black", strokeWidth: 4}), ] }); return html`

${plot.outerHTML}

An input of ${inp} produces ${numSpikes} spike${numSpikes===1 ?'' : 's'} in ${runtime} seconds (${numSpikes / runtime} Hz)

`; }4b:T5e7,{ const lif = new LIF(); const alif = new ALIF(); alif.tau_inh = tau_inh2; alif.inc_inh = inc_inh2; // alif.ref = lif.ref = 0; const INPUT_STEP = 1e-2; const [INPUT_MIN, INPUT_MAX] = [0.9, 2]; const rates = []; for(let inp = INPUT_MIN; inp <= INPUT_MAX; inp += INPUT_STEP) { const lifRate = getFiringRate(inp, lif); const alifRate = getFiringRate(inp, alif); const analyticalLIFRate = (inp <= lif.v_th) ? 0 : (1 / (lif.tau_ref - lif.tau_rc * Math.log(1 - lif.v_th/inp))); // const analyticalALIFRate = (inp <= lif.thresh) ? 0 : (1 / (alif.ref - alif.rc * Math.log(1 - alif.thresh/inp)) - (alif.tau_inh) * Math.log(alif.inc_inh)); // const analyticalALIFRate = (inp <= alif.thresh) ? 0 : (1/(alif.ref + Math.log(z(inp - alif.thresh)/(inp * Math.exp(-1/alif.rc))))); // reset state variables so that this sim does not have an effect on the next one lif.v = alif.v = 0; lif.refractory_time = alif.refractory_time = 0; alif.inh = 0; rates.push({ lifRate, alifRate, inp, analyticalLIFRate }); } const plot = Plot.plot({ y: { grid: true, label: 'j' }, marks: [ Plot.line(rates, {x: "inp", y: "alifRate", stroke: "red"}), Plot.line(rates, {x: "inp", y: "lifRate", stroke: "black"}), Plot.line(rates, {x: "inp", y: "analyticalLIFRate", stroke: "grey", strokeWidth: 0.5}), // Plot.line(rates, {x: "inp", y: "analyticalALIFRate", stroke: "blue", strokeWidth: 0.5}), ] }); return plot; }4c:[] 46:{"id":"9f9286d1e917416a","slug":"nmc-adaptive-lif-neurons","title":"Adaptive LIF Neurons","thumbnail":"a5fe4066b11fc55004b204dfb1d61d66883d3cd8999ce91a02a6ba92a745a05f","access_level":"public","create_time":"$D2022-11-28T13:42:20.185Z","update_time":"$D2023-02-21T21:12:39.901Z","view_time":null,"first_public_time":"$D2023-01-24T15:22:14.845Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"55b87b37da07de37","login":"soney","name":"Steve Oney","avatar_url":"https://avatars-next.observableusercontent.com/565926373d73cf99aa6425cf7c0d7ff79e41635d1a367894a5e3bce0299b18b9","tier":"starter_2024","bio":"","following":false},"version":489,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"1d138075b46297f1","login":"soney","name":"Steve Oney","avatar_url":"https://avatars-next.observableusercontent.com/565926373d73cf99aa6425cf7c0d7ff79e41635d1a367894a5e3bce0299b18b9","tier":null,"bio":"","following":false}],"members":[{"type":"individual","id":"79581e18d086caf4","login":"ericgallo","name":"","avatar_url":null,"tier":null,"role":"editor"},{"type":"individual","id":"93113dd5fb467076","login":"sharvari-bondre","name":"Sharvari Bondre","avatar_url":"https://avatars-next.observableusercontent.com/754b967a5cf1ce71589f615fed536b94bd68509f3412154d82572c4b6aa44ead","tier":null,"role":"editor"},{"type":"individual","id":"1d138075b46297f1","login":"soney","name":"Steve Oney","avatar_url":"https://avatars-next.observableusercontent.com/565926373d73cf99aa6425cf7c0d7ff79e41635d1a367894a5e3bce0299b18b9","tier":null,"role":"editor"}],"comments":[{"id":"ccb97b56d5b17e55","reply_to_id":null,"content":"Analytical solution to ALIF tuning curve?","create_time":"$D2023-01-11T20:44:35.118Z","update_time":null,"resolved":false,"user":{"type":"individual","id":"1d138075b46297f1","login":"soney","name":"Steve Oney","avatar_url":"https://avatars-next.observableusercontent.com/565926373d73cf99aa6425cf7c0d7ff79e41635d1a367894a5e3bce0299b18b9","tier":null}}],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"ceab76d8d8a0acbc","slug":"neuromorphic-computing","title":"Neuromorphic Tutorials: Spiking Neural Networks (SNNs) from Scratch","owner":{"id":"55b87b37da07de37","login":"soney","name":"Steve Oney"}},"type":"notebook","body":{"titling":"auto","title":"Adaptive LIF Neurons","cells":[{"id":0,"value":"# Adaptive LIF Neurons","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"Thusfar, in modeling leaky integrate-and-fire (LIF) neurons, the only state variable that we have been modifying has been its current voltage/potential. However, in some applications it can also be useful to allow the **threshold voltage** (${tex`v_{th}`}) to vary. Specifically, when there is a spike, we can slightly increase ${tex`v_{th}`} so that it is slightly more difficult to spike next time (i.e., it adapts to the level of input). We will call our new neuron model \"Adaptive LIF\" (ALIF).\n\nThe ALIF can be useful in a variety of scenarios, including cases where we want neurons to \"fine tune\" their behave to adapt to input that might otherwise be too \"intense\" (and thus result in too high of a firing rate if neurons don't adapt) and cases where we want to be sure that one neuron in a group doesn't get \"greedy\" for inputs with which it is closely aligned.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":27,"value":"To implement ALIF neurons, we will create an additional state variable ${tex`w`} to represent how much the threshold voltage (${tex`v_{th}`}) has increased (meaning that we will spike only when ${tex`v \\geq v_{th} + w`}). ${tex`w`} will start as ${tex`0`} and increase every time our neuron fires. We also create a variable ${tex`b`} to represent the **strength of adaption**. When our ALIF spikes, we will increment ${tex`w`} by ${tex`b`}. We will also introduce a time constant ${tex`\\tau_w`} to specify how quickly ${tex`w`} decays back to ${tex`0`}.\n\nSo now, rather than a static threshold ${tex`v_{th}`}, we have a dynamic threshold, which we will call ${tex`\\vartheta_{th}(t)`}:\n\n${tex`\\vartheta_{th}(t) = v_{th} + w(t)`}\n\nand our threshold changes over time by:\n\n${tex`\\vartheta_{th}'(t) = -\\frac{1}{\\tau_w}w(t)`}\n\nif there is a spike at time ${tex`t`} then ${tex`w(t)`} increases by ${tex`b`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":18,"value":"To implement ALIF neurons, let's start by recalling the code that we had for LIF neurons:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":21,"value":"class LIF {\n constructor(tau_rc=0.2, v_init=0, tau_ref=0.002, v_th=1){\n this.tau_rc = tau_rc;\n this.tau_ref = tau_ref;\n this.v = v_init;\n this.v_th = v_th;\n \n this.output = 0;\n this.refractory_time = 0;\n }\n\n step(I, t_step) {\n this.refractory_time -= t_step; // Subtract the amount of time thhat passed from our refractory time\n \n if(this.refractory_time < 0) { // If we aren't in our refractory period, update the voltage\n this.v = this.v * (1 - t_step/this.tau_rc) + (I * t_step / this.tau_rc);\n }\n\n if(this.v >= this.v_th) { // Fired\n this.refractory_time = this.tau_ref; // Set refractory time\n\n this.output = 1 / t_step; // Fire\n this.v = 0; // Reset potential\n } else {\n this.output = 0; // Don't fire\n }\n return this.output;\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"$48","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":64,"value":"$49","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":72,"value":"$4a","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":174,"value":"viewof tau_inh = Inputs.range([0, 1], {step: 0.001, label: \"tau_inh\", value: 0.3})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":176,"value":"viewof inc_inh = Inputs.range([0, 2], {step: 0.001, label: \"inc_inh\", value: 0.2})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":226,"value":"t_short = 0.55;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":205,"value":"Let's see what this looks like with `tau_inh = ${tau_inh}` and `inc_inh = ${inc_inh}` over a short time period (${t_short} seconds). Note that we reduced the neuron's refractory period so that the effect is more visible:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":74,"value":"plotSpikes(1.3, tau_inh, inc_inh, t_short)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":231,"value":"t_long = 5;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":213,"value":"Note that at first, the neuron spikes quickly but then spikes become more spaced out over time. The firing speed does become more regular over the long term, as our threshold and voltage \"balance\" out. Let's look over a period of ${t_long} seconds:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":220,"value":"plotSpikes(1.3, tau_inh, inc_inh, t_long)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":243,"value":"Finally, we claimed that part of the motivation for using ALIFs is that their firing rate can be less sensitive to input variability...so let's test this. Again, to make the difference more visible, we are going to tune a parameter of our ALIF neurons...we are going to increase `tau_inh` to `${tau_inh2}` so that the additional inhibition stays for \"longer\" after it is increased.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":247,"value":"function getFiringRate(inp, neuron, runtime=10) {\n const TIME_STEP = 1e-3;\n const NUM_STEPS = runtime / TIME_STEP; // Simulate for runtime seconds\n const J = () => inp;\n let numSpikes = 0;\n for(let i = 0; i < NUM_STEPS; i++) {\n const j = J((i+1)*TIME_STEP);\n neuron.step(j, TIME_STEP);\n const out = neuron.output;\n if(out > 0) {\n numSpikes++;\n }\n }\n return numSpikes / runtime;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":273,"value":"viewof tau_inh2 = Inputs.range([0, 1], {step: 0.001, label: \"tau_inh\", value: 0.7})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":311,"value":"viewof inc_inh2 = Inputs.range([0, 20], {step: 0.001, label: \"inc_inh\", value: 1})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":254,"value":"$4b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":282,"value":"Note here how the ALIF (red line) is less sensitive to changes in input than the normal LIF. If we decrease `tau_inh`, our ALIF behaves more and more like the LIF model (since the inhibition disappears more quickly). Similarly, if we decrease `inc_inh`, our ALIF behaves more and more like the LIF model (since the inhibition changes less).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":464,"value":"# Summary\n\n- Adaptive LIF neurons introduce a variable firing threshold ${tex`\\vartheta_{th}(t) = v_{th} + w(t)`}\n - ${tex`v_{th}`} is the \"original\" firing threshold\n - ${tex`w_{t}`} is an additional threshold that changes over time: ${tex`w'(t) = -\\frac{1}{\\tau_w}w(t)`}\n - ${tex`\\tau_w`} is a time constant that controls how quickly ${tex`w(t)`} decays\n - if there is a spike at time ${tex`t`} then ${tex`w(t)`} increases by some constant ${tex`b`}.\n- Adaptive LIFs can help us make our LIFs' firing rate less sensitive to input variability","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":456,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":478,"value":"# Adding Precision (Optional)\n\nAgain, we used a first-order estimate to update our value of ${tex`w(t)`}:\n\n```JavaScript\nthis.inh = this.inh * (1 - t_step/this.tau_inh);\n```\n\nJust as was the case with [LIFs](https://observablehq.com/@soney/modeling-lif-neurons?collection=@soney/neuromorphic-computing#cell-87) and [synapse models](https://observablehq.com/d/3c638d3aaf9207b4?collection=@soney/neuromorphic-computing#cell-376), we can get a better estimate by analytically modeling ${tex`w[t]`} as ${tex`w[t] = w[t-1]e^{-\\frac{T_{step}}{\\tau_w}}`}:\n\n```JavaScript\nthis.inh = this.inh * Math.exp(-t_step/this.tau_inh);\n```","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"}],"files":"$W4c","stdlib":"1"}} 47:"$46:body"