# A Black-box Model for Neurons

TL;DR: It is shown that, after training the artificial network with biologically plausible input currents, the network is able to identify the neuron’s behaviour with high accuracy, thus obtaining a black box that can be then used for predictive goals.

Abstract: We explore the identification of neuronal voltage traces by artificial neural networks based on wavelets (Wavenet). More precisely, we apply a modification in the representation of dynamical systems by Wavenet which decreases the number of used functions; this approach combines localized and global scope functions (unlike Wavenet, which uses localized functions only). As a proof-of-concept, we focus on the identification of voltage traces obtained by simulation of a paradigmatic neuron model, the Morris-Lecar model. We show that, after training our artificial network with biologically plausible input currents, the network is able to identify the neuron’s behaviour with high accuracy, thus obtaining a black box that can be then used for predictive goals. Interestingly, the interval of input currents used for training, ranging from stimuli for which the neuron is quiescent to stimuli that elicit spikes, shows the ability of our network to identify abrupt changes in the bifurcation diagram, from almost linear input-output relationships to highly nonlinear ones. These findings open new avenues to investigate the identification of other neuron models and to provide heuristic models for real neurons by stimulating them in closed-loop experiments, that is, using the dynamic-clamp, a well-known electrophysiology technique.

## Summary (1 min read)

### Introduction

- Neurons are the basic information processing structures in the brain.
- There is a vast literature on modeling of such intrinsic features, see for instance [3] for a thorough treatment.
- A plethora of experiments has since been devoted to provide specific models by identifying and quantifying the ionic channels, giving rise to very precise biophysical models now available to the computational neuroscience community.
- Thus, the problem of identification and cell classification from voltage traces is fundamental to experimental neuroscience, see [5] where the problem of detection, time-estimation, and cell classification is treated in order to sort neural action potentials.
- The identification method decreases the number of used functions in Wavenet by combining localized and global scope functions instead of only localized functions.

### III. THE NETWORK CHARACTERISTICS

- Great advances were made in the last years in analysis and identification of dynamical systems using non-linear models originated from artificial intelligence.
- Mathematically, they are complex models, whose structure is empirically determined.
- Network structure parameters and training method are determined by error and trial or Heuristic.
- In the multiresolution frame, the approximation of a function f(x) is made through its projections to shifted and compressed versions of a basic function, known as “wavelet mother”.
- Training data are initially approximated with activation functions (scale functions), whose support is equal to the problem domain support (global scope functions), different from the originally proposed wavenet, which uses localized functions only.

### A. Dynamical system identification

- The authors deal with the identification of the neuronal voltage traces of the Morris-Lecar model proposed in [1].
- The steps followed in the identification process were: 1) Acquisition of data group for fitting (Training Patterns): data were obtained solving system 2.
- As a measure criterion, the smaller quadratic error with the smaller number of variables was considered.
- 3) The validation trough dynamic prediction, which corresponds to the prediction of an arbitrary number of steps forward.
- In relation to the other points, only the information of the perturbation variable is used, as external information, and a feedback of the output variables is performed.

### B. Simulation results

- The neural network was trained by defining the Iapp current as an independent variable.
- Iapp is defined as a piecewise constant signal with 50 levels randomly defined with a uniform distribution.
- The value of the constant changes every 2000 integration steps .
- The solutions of the differential equations the model and the neural network prediction are depicted (actually overlapped) in the next figures.
- As it can be verified from the results presented, the prediction in both subthreshold and trigger conditions is satisfactory.

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##### References

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...Nevertheless, systems identification can be very tiring, due to the great number of network structure parameters (number of hidden layers, number of neurons per layer) and training method (weights, initial selection, learning factor determination, moment rate and stopping criteria) [7]....

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