NEURAL ENCODING KAI LEI AND STEPHANIE WOROBEY Math 390 Mathematical Biology, Fall 2015 Advisor: Viktor Grigoryan Simmons College Introduction There are approximately 1000 billion neurons in a human body, and each one of them are essential to our perceptions of senses. The re- sponses initiated by internal and external stimulus enable us to see, hear, smell, taste and touch. The major function of neurons is to transmit information throughout the nervous system. They achieve this by generating electrical pulses called action potentials or spikes and these spikes appear in various patterns. Neural coding examines upon how action potentials represent stim- ulus attributes and this can be done from two opposite perspectives: neural encoding or neural decoding. In particular, we are looking at neural encoding, which is the map from stimulus to response. Before introducing the algorithms that model different aspects of neural en- coding, we will go over the fundamental mechanism of how neurons communicate with each other and also the basic structures of a neuron and their functions. Properties of a Neuron and their Functions [2] Date: Fall 2015. 1 2 KAI LEI AND STEPHANIE WOROBEY The cell body of a neuron is responsible for the biological machinery to keep the cell alive. Dendrites are fibers that project out of the cell body and they collect signals from other neurons. Signals can be classified as either excitatory or inhibitory. [2] Excitatory signals tells the neuron to generate an electrical impulse whereas inhibitory signals tells the neuron not to generate an electrical impulses. When many excitatory signals are received, the neuron reaches threshold of activation and an electrical impulse also known as action potential is generated and would travel down the axon to the axon terminals. The synapse also known as the synaptic gap is a small space between the axon terminals of one neuron and the dendrites of another neuron. The axon terminal contains sacs of neurotransmitters which are nat- urally occurring chemicals that specialize in transmitting information between neurons [2]. Once the action potential reached the axon ter- minals, the neurotransmitter is then released and proceeds to bind to the receptors of the receiving neuron. Next, we introduce methods of recording neuronal responses. How to Record Neuronal Responses There are two ways to record action potentials electrically : either intracellularly or extracellulary. The extracellular method involves first connecting a neuron to electrolyte filled, glass electrode. There are then two ways to record the readings. The first is by inserting a sharp electrode into the cell. The second, by sealing patch electrodes to the surface of the membrane of the cell. The seal of the patch electrode causes the membrane to break, and then the tip of the patch electrode NEURAL ENCODING 3 [3] Figure 1. These are three simulated responses from a neuron. The top one shows an intracellular electrode reading from the soma. The bottom reading shows an intracellular reading from an electrode connected to the axon. The middle trace is an extracellular reading, in which no subthreshold potentials are present. has access to the interior of the cell. It is observed that the subthresh- old membrane potentials can be seen in the soma of neurons, but is not observable in the axon { therefore spikes, but not subthreshold potentials reproduce regeneratively down axons. In extracellular readings, the electrode never pierces the membrane of the cell. The electrode is just placed near the neuron. These kinds of recordings, however, can only record action potentials, and they are incapable of recording subthreshold potentials. Complication of Neural Coding It is often difficult to demonstrate the relationship between stimulus and response because neuronal responses can vary significantly trail from trial even if when the stimulus that's presented is the same dur- ing every trial. The potential factors that vary the responses includes levels of arousal or attention, effects of various biophysical or cognitive processes, etc. For example, when the same person brushes one's arm 4 KAI LEI AND STEPHANIE WOROBEY repeatedly at the same spot, the person being brushed might still feel different every time. This complexity of variations contributes to the unlikelihood of de- termining and predicting when an action potential would occur. And the model we are presenting below "accounts for the probabilities that different spike sequences are evoked by a specific stimulus" [3]. In other words, we are presenting a probabilistic model that would counteract the stochastic nature of neuronal responses. Also we need to take in account of the fact that whenever a stim- ulus is presented, there are usually more than one neuron that would respond. Therefore, aside from investigating the firing pattern of one particular neuron, we also need to look at how these firing patterns relate to each other. Firing Rates The first function in this model is the neural response function. This function is constructed under 3 assumptions. The first assumption is that even though action potentials acts differently depending on their duration, amplitude and shape, we are going to treat them as iden- tical stereotyped event. The second assumption is that an action potential sequence can be represented by a list of the times when the spikes occurred because the timing determines how and when a spike transmits information. So for n number of spikes, we use the notation ti to represent the spike times. The last assumption is that during each trial that the spikes are recorded, we start at time 0 and end at time T , so that puts ti in the interval between 0 and T , inclusively. Based on these 3 assumptions, the spike sequence is represented as a sum of idealized spikes using the Delta function as followed: n X (1) ρ(t) = δ(t − ti) i=1 Delta function is a generalized function on the real number line that is zero everywhere, except at 0, with an integral of one over the entire real line. This function is used because when we model it, its physical nature mirrors what a spike would look like- "it is sometimes thought of as an infinitely high, infinitely thin spike at the origin". ρ(t) is the neural response function and we can use it to re-express sums over spikes as integrals over time. For well-behaved function h(t), we have the following function: NEURAL ENCODING 5 n X Z T (2) h(t − ti) = h(τ)ρ(t − τ)dτ i=1 0 where the integral is expressed over the length of the trial. Using the basic definition of a δ function, we get: Z (3) δ(t − τ)h(τ)dτ = h(t) provided that the limits of the integral surround the point t and if they do not, the integral is zero. Well behaved functions are functions that do not violate the 3 assumptions mentioned above. Neuronal responses are treated probabilistically because action po- tentials generated by the same stimulus can vary trial from trial. If we seek the probability for a spike to occur at any specified time, we would get a 0 value because spike times are continuous variables. Instead, we seek the probability for a spike to occur over a specified time interval between time t and t + ∆t. Furthermore, we use the notation P [ ] to represent probabilities and p[ ] to represent probability densities. We use angle brackets, hi to represent average over trials given the same stimulus. Applying these notations, we use p[t]∆t to represent the probability that a spike occurs between times t and t + ∆t, where p[t] is the single spike probability density. The quantity p[t] can also be defined as the firing rate of the cell, and we use r(t) to denote it. A way to approximate r(t) is to determine the fraction of trials with a given stimulus on which a spike occured between the times t and t + ∆t. For small ∆t and large numbers of trials, this method produces a good approximation under the Law of Large Numbers, which states that: as the number of trials gets larger, the relative frequency approximation of P(A) gets closer to the theoretical value. [1] Next, we use hρ(t)i to represent the trial-averaged neural response function, which produces the fraction of trails on which a spike occurs. Using this relationship, we get the following: Z t+∆t (4) r(t)∆t = hρ(t)idτ t And for well behaved functions h, we replace hρ(t)i with r(t) and we have: 6 KAI LEI AND STEPHANIE WOROBEY Z Z (5) h(τ)hρ(t − τ)idτ = h(τ)r(t − τ)dτ This function is important because it demonstrates the relationship between hρ(t)i and r(t). Another important quantity in neural encoding is the spike-count rate r and we get this value by "counting the number of action poten- tials that appear during a trial and divide by the duration of the trial". [3] n 1 Z T (6) r = = ρ(τ)dτ T T 0 The spike-count rate r can also be defined as the time average of ρ(t) over the duration of the trial. The average firing rate can be obtained by averaging r(t) over trials and we get: hni 1 Z T 1 Z T (7) hri = = hρ(τ)idτ = r(t)dt T T 0 T 0 Measuring Firing Rates: Linear Filter and Filter Kernel The Linear Filter, and the Filter Kernel (or Window Function) are two ways to approximate r(t), or the firing rate. The image below is of the firing rate approximated by sliding the rectangular window function along the spike train, where ∆t is 100 ms.