Notes

Motivation Neurobiology mostly uses spiking neural networks. Neurons output spikes, which are binary events and localized in time. So how do hidden units learn?

Bottom-Up Approach:

• Start with a random network model.
• Include data driven plasticity model.
• Observe function Limited success in learning useful hidden layer representations.

One outcome would be Spike-Time Dependent Plasticity (STDP), which is a way the weights can be adjusted. So far, not very useful in building networks; the weights tend to blow up. Over the years, people went over this concept and tried to improve.

Top-Down Approach:

• Start with a function in mind.
• Derive suitable plasticity rules.
• Build functional network models.

Deep Learning is an example of a top-down framework. Two questions remain:

• The Algorithmic Question: How to compute the gradient?
• The Conceptual Question: Which functions are learned?

Recap: Spiking Neuron Models

• Spiking Networks consist of Spiking Neurons.
• Network Modelling largely relies on Simplified Neuron Models.

Biophysics of Neuronal Signal Transmission

From Biophysical to Reduced Neuron Models In order to build a phenomenological model of neuronal dynamics, we describe the critical voltage for spike initiation by a formal threshold $\theta$. If the voltage $U_{i}(t)$ (that contains the summed effect of all inputs) reaches $\theta$ from below, we say that neuron i fires a spike. The moment of threshold crossing defines the firing rate $t_{i}^{(f)}$. The models makes use of the fact that neuronal action potentials of a given neuron always have roughly the same form. If the shape of an action potential is always the same, then the shape cannot be used to transmit information: rather information is contained in the presence or absence of a spike. Therefore action potentials are reduced to "events" that happen at a precise moment in time.

Leaky Integrate-and-Fire Neuron Neuron models where action potentials are described as events are called "Integrate-and-Fire" models. No attempt is made to describe the shape of an action potential. Integrate-and-Fire models have two separate components that are both necessary to define their dynamics:

1. An equation that describes the evolution of the membrane potential $U_{i}(t)$.
2. A mechanism to generate spikes.

The variable $U_{i}$ describes the momentary value of the membrane potential of neuron i. In the absence of any input, the potential is at its resting value $U_{rest}$. If an experimentalist injects a current $I(t)$ into the neuron, or if the neuron receives synaptic input from other neurons, the potential $U_{i}(t)$ will be deflected from its resting value. The basic electrical circuit representing a leaky integrate-and-fire model consists of a capacitor C in parallel with a resistor R driven by a current $I(t)$, as shown in the figure below. The differential equation for describing the leaky-integration of the voltage is given by:

where $\tau = RC$ is the time constant of the circuit.

Now, the second part of the leaky integrate-and-fire neuron is the firing and re-setting of the voltage after the neuron-specific threshold has been reached. At the firing time: $t^{f}:U\left( t^{f} \right) = \ \theta$, the neuron fires (with a not-here-to-be-defined spike-form), the firing time is noted and immediately after the voltage reset to a new value $U_{rest} < \ \theta$:

Exponential Postsynaptic Currents

We also want to model the synapse. Activation of a presynaptic neuron results in a release of neurotransmitters into the synaptic cleft. The transmitter molecules diffuse to the other side of the cleft and activate receptors that are located in the postsynaptic membrane. In both cases, the activation of the receptor results in the opening of certain ion channels and, thus, in an excitatory or inhibitory postsynaptic transmembrane current (EPSC or IPSC). The main mechanism for carrier transport underlying this current is diffusion of the ions passing from the extracellular space into the cell. Instead of developing a mathematical model of the transmitter concentration in the synaptic cleft, we keep things simple and describe transmitter-activated ion channels as an explicitly time-dependent conductivity. This conductivity change most often modelled as an exponentially decaying unction, to represent the effect of closing ion channels. The following differential equation describes the evolution of the postsynaptic current $I(t)$:

Considering the changes arising due to the discrete APs, the term S(t), added in the equation, determines an instantaneous increase in the postsynaptic current proportional to the synaptic weight.

The Spike Response Model (SRM0) So far, we have described neuronal dynamics in terms of systems of differential equations. There is another approach called the "filter picture". In this picture, the parameters of the model are replaced by (parametric) functions of time, generically called "filters". The neuron model is therefore interpreted in terms of a membrane filter as well as a function describing the shape of the spike and, potentially, also a function for the time course of the threshold. Together, these three functions establish the Spike Response Model (SRM). Mathematically speaking, we integrate over the differential equation, then replace the integration times multiplications with convolutions of filter kernels over the spikes: