You mean the temporal derivative of an input waveform? Or something else?
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Replying to @AdamMarblestone @KordingLab and
I am interested in a general process for dendritic integration where the synaptic inputs are the function values f_i of a function of several variables and the output of the dendritic computation is the partial derivative of f_i with respect to one or more variables.
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Replying to @bayesianbrain @AdamMarblestone and
Aidan Rocke Retweeted KordingLab 👨💻 🧠∇ 🔬 📈, 🏋️♂️ ⛷️ 🏂 🛹 🕺 ⛰️ ☕ 🦖
In general, this may be a function learned by a network of neurons. So I think this definition of
@KordingLab might be a special case: https://twitter.com/KordingLab/status/1220101640432254976 … If spike trains can encode ordered pairs (x,f(x))...it should be possible.Aidan Rocke added,
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Replying to @bayesianbrain @KordingLab and
Still too abstract for me — need a concrete example say a fxn of 2 variables.
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Replying to @AdamMarblestone @KordingLab and
For concreteness, I think dendritic trees may compute the partial derivatives of functions using an algorithm similar to the Cauchy Integral Formula for derivatives. It can be implemented on any binary tree where a large number of local computations occur in parallel.
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Replying to @bayesianbrain @AdamMarblestone and
For a large number of functions(ex. polynomials, sigmoid, trigonometric) we have geometric convergence in error. I think any proposed algorithm must scale very well with the dimension of the input to the dendritic tree. Very fast convergence implies robustness.
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Replying to @bayesianbrain @KordingLab and
Like say I am a synapse, the jth in this tree. Function f(x,y). I have an associated x_j, y_j and my input at time t is taken as f(x_j, y_j)? Now what do you want the neuron spike output to represent? Or is what Konrad said the thing you want?
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Replying to @AdamMarblestone @KordingLab and
If we represent these inputs as a vector [f(x_1,y_1),...,f(x_n,y_n)] then the spike output may represent the dot product of [f(x_1,y_1),...,f(x_n,y_n)] and [f_1,...,f_n] where the f_i denote frequencies of different voltage oscillations in a dendritic tree.
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Replying to @bayesianbrain @AdamMarblestone and
This formulation is not exact in the sense that it's not Contour Integration but the information I need is there. I think that by averaging these computations we can get a good stochastic estimate of a partial derivative with respect to any x_i or y_i.
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Ok so the partial derivative with respect to either x or y at any of those points (x_j, y_j)? I don’t know the answer but I think I get the question!
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Replying to @AdamMarblestone @bayesianbrain and
Or just at one point x_0, y_0?
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Replying to @AdamMarblestone @bayesianbrain and
I think I can very loosely see from your contour integration and argument about summing oscillations how you think you might be able to do the latter.
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