Question

READ THIS WELL: If you answer with something stupid/inccorect (e.g. telling me what a depolarization is or telling me that a small postsynaptic cell helps a signal carry, I'll give you no credit)

One property of electrical synapses is that the postsynaptic neuron tends to be smaller than the presynaptic neuron. Based on what you know about the relationship between neuron size and input resistance, what is the advantage of this arrangement? Please answer this in terms of Ohm's law considering input resistance and delta voltage. I'm not sure if I should say in my answer that a greater delta-v (which corresponds to a greater input resistance seen in smaller cells) causes the action potential to actually OCCUR in the postsynaptic cell, or if it just makes the action potential happen QUICKER? is the rate of propagation changed or is there just more likely that an actual action potential will happen (with a bigger postsynaptic cell than presynaptic cell)?

Answer 1

As a convention, the neuron transmitting or generating a spike and incident onto a synapse is referred as the presynaptic neuron, whereas the neuron receiving the spike from the synapse is referred as the postsynaptic neuron (see Figure 2.3). Also, there are two types of synapses typically encountered in neurobiology: excitatory synapses and inhibitory synapses. For excitatory synapses, the membrane potential of the postsynaptic neuron (referred to as the excitatory postsynaptic potential, or EPSP) increases, whereas for inhibitory synapses, the membrane potential of the post-synaptic neuron (referred to as the inhibitory postsynaptic potential, or IPSP) decreases.It is important to note that the underlying dynamics of EPSP, IPSP, and the action potential are complex and several texts have been dedicated to discuss the underlying mathematics. Therefore, for the sake of brevity, we only describe a simple integrate-and-fire neuron model that has been extensively used for the design of neuromorphic sensors [9] and is sufficient to explain the noise exploitation techniques described in this chapter.

We first define a spike train ρ(t) using a sequence of time-shifted Kronecker delta functions as

(2.1)ρ(t)=∑m=1∞δ(t-tm),

where δ(t) = 0 for t ≠ 0 and ∫-∞+∞δ(τ)dτ=1. In the above Eq. (2.1), the spike is generated when t is equal to the firing time of the neuron t_m.If the somatic (or membrane) potential of the neuron is denoted by v(t), then the dynamics of the integrate-and-fire model can be summarized using the following first-order differential equation:

(2.2)ddtv(t)=-v(t)/τm-∑j=1NWj[h(t)∗ρj(t)]+x(t),

where N denotes the number of presynaptic neurons, W_j is a scalar transconductance representing the strength of the synaptic connection between the jth presynaptic neuron and the postsynaptic neuron, τ_m is the time constant that determines the maximum firing rate, h(t) is a presynaptic filtering function that filters the spike train ρ_j(t) before it is integrated at the soma, and * denotes a convolution operator. The variable x(t) in Eq. (2.2) denotes an extrinsic contribution to the membrane current, which could be an external stimulation current. When the membrane potential v(t) reaches a certain threshold, the neuron generates a spike or a train of spikes. Again, different chaotic models have been proposed that can capture different types of spike dynamics. We next briefly describe different methods by which neuronal spikes encode information.

The simplest form of neural coding is the rate-based encoding that computes the instantaneous spiking rate of the ith neuron R_i(t) according to

(2.3)Ri(t)=1T∫tt+Tρi(t)dt,

where ρ_i(t) denotes the spike train generated by the ith neuron and is given by Eq. (2.1), and T is the observation interval over which the integral or spike count is computed. Note that the instantaneous spiking rate R(t) does not capture any information related to the relative phase of the individual spikes, and hence it embeds significant redundancy in encoding. However, at the sensory layer, this redundancy plays a critical role because the stimuli need to be precisely encoded and the encoding have to be robust to the loss or temporal variability of the individual spikes.

Another mechanism by which neurons improve reliability and transmission of spikes is through the use of bursting, which refers to trains of repetitive spikes followed by periods of silence. This method of encoding has been shown to improve the reliability of information transmission across unreliable synapses  and, in some cases, to enhance the SNR of the encoded signal. Modulating the bursting pattern also provides the neuron with more ways to encode different properties of the stimulus. For instance, in the case of the electric fish, a change in bursting signifies a change in the states (or modes) of the input stimuli, which could distinguish different types of prey in the fish’s environment

Whether bursting is used or not, the main disadvantage of rate-based encoding is that it is intrinsically slow. The averaging operation in Eq. (2.3) requires that a sufficient number of spikes be generated within T to reliably compute R_i(t). One possible approach to improve the reliability of rate-based encoding is to compute the rate across a population of neurons where each neuron is encoding the same stimuli. The corresponding rate metric, also known as the population rate R(t), is computed as

(2.4)R(t)=1N∑i=1NRi(t),

where N denotes the number of neurons in the population. By using the population rate, the stimuli can now be effectively encoded at a signal-to-noise ratio that is N^1/2 times higher than that of a single neuron [15]. Unfortunately, even an improvement by a factor of N is not efficient enough to encode fast-varying sensory stimuli in real time. Later, we show that lateral inhibition between the neurons would potentially be beneficial to enhance the SNR of a population code by a factor of N²[16] through the use of noise shaping.

We complete the discussion of neural encoding by describing other forms of codes: time-to-first spike, phase encoding, and neural correlations and synchrony. We do not describe the mathematical models for these codes but illustrate the codes using Figure 2.4d.

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Figure 2.4. Different types of neural coding: (a) rate, (b) population rate, (c) burst coding, (d) time-to-spike pulse code, (e) phase pulse code, and (f) correlation and synchrony-based code. Adapted from Ref. 13.

The time-to-spike is defined as the time difference between the onset of the stimuli and the time when a neuron produces the first spike. The time difference is inversely proportional to the strength of the stimulus and can efficiently encode the real-time stimuli compared to the rate-based code. Time-to-spike code is efficient since most of the information is conveyed during the first 20–50 ms [17, 18]. However, time-to-first-spike encoding is susceptible to channel noise and spike loss; therefore, this type of encoding is typically observed in the cortex, where the spiking rate could be as low as one spike per second.

An extension of the time-to-spike code is the phase code that is applicable for a periodic stimulus. An example of phase encoding is shown in Figure 2.4e, where the spiking rate is shown to vary with the phase of the input stimulus. Yet another kind of neural code that has attracted significant interest from the neuroscience community uses the information encoded by correlated and synchronous firings between groups of neurons. The response is referred to as synchrony and is illustrated in Figure 2.4f, where a sequence of spikes generated by neuron 1, followed by neuron 2 and neuron 3, encodes a specific feature of the input stimulus. Thus information is encoded in the trajectory of the spike pattern and so can provide a more elaborate mechanism of encoding different stimuli and its properties