AMATH Quiz Practice

What is the firing rate model equation from class and homework?

τ dri/dt = -ri + f(inputsi) where ri is the firing rate of neuron i τ is the time constant and f is the function relating input to steady-state firing rate.

What does r_i represent in the firing rate model equation?

r_i represents the firing rate of cell i.

What does the function f represent in the firing rate model equation?

The function f represents the steady state firing rate that neuron i will achieve in response to inputs_i like the frequency-input curve plotted for the Hodgkin Huxley equations.

If the term inputsi is constant in time which best describes the evolution of ri over time?

ri will approach f(inputsi) over time approaching the steady state firing rate determined by the constant input.

What does the term Wij represent in the firing rate model equation where inputsi = Σj Wij rj(t) + Ii(t)?

W_ij represents the synaptic weight from cell j to cell i.

What choices of the function f would always result in the type of linear firing rate equations studied in the first part of the homework?

f(inputs) = inputs (linear) and f(inputs) = RELU(inputs) (rectified linear) both result in linear firing rate equations with RELU being linear above threshold.

What eigenvalue best describes selective amplification of the stimulus in the direction of the corresponding eigenvector?

The eigenvalue with real part closest to 1 results in selective amplification of the stimulus in the direction of the corresponding eigenvector.

What is the Erdos-Reyni model of a random graph?

The Erdos-Reyni model has all connections made at random and independently with a given probability p for each possible edge.

What does the Hodgkin-Huxley equations contain in terms of conductances?

The Hodgkin-Huxley equations contain a mix of voltage-dependent conductances and voltage-independent conductances including sodium potassium and leak channels.

For HIGHER values of the constant potassium conductance gK extra added in class what typically happens?

Higher values of constant potassium conductance gK extra have a suppressive effect on firing rates causing the model neuron to fire less or stop firing.

For HIGHER values of the Calcium-dependent potassium conductance added in the homework what typically happens?

For higher values of the Calcium-dependent potassium conductance more positive charge (current) flows OUT of the neuron causing hyperpolarization and reduced firing.

What is the role of synaptic depression in the model from homework and class for constant input rates at steady state?

Synaptic depression makes the product of the synaptic release probability Prel and the incoming spike rate approximately independent of the input rate providing gain control.

The RC circuit model with current-based input and initial voltage V(0) = 0 exactly follows which principle?

The RC circuit model with current-based input and initial voltage V(0) = 0 exactly follows the superposition principle where response to sum of inputs equals sum of individual responses.

Suppose x(t) solves dx/dt = f(x) with initial condition x(0) = 0. What is the Euler method solution for x at time 1 with timestep Δt = 1?

The Euler method solution for x at time 1 with timestep Δt = 1 is Δt * f(0) which equals f(0).

What is the primary biological feature of a neuron that gives rise to the term g in the effective neuron model?

The primary biological feature giving rise to the term g is the ion channel conductance.

What is the primary biological feature of a neuron that gives rise to the term C in the RC circuit model?

The primary biological feature giving rise to the term C is the insulating properties of the cell membrane which acts as a capacitor.

What is the Fano factor of an ideal mathematical Poisson process?

The Fano factor of an ideal mathematical Poisson process is 1 where variance equals mean.

A spike count generated by a Poisson process with constant firing rate that is the same on each trial will have zero variance where variance is measured from trial to trial?

False. A Poisson process has variance equal to its mean so there is trial-to-trial variability.

When defining the encoding property of a cell which probability distribution is most related?

Encoding is most related to P(response | stimulus) describing how stimuli cause patterns of responses.

When defining the decoding property of a cell which probability distribution are we defining?

Decoding is defining P(stimulus | response) describing how to infer stimuli from responses.

Consider two Poisson processes: Process 1 constant rate Process 2 time-varying rate. What can we conclude about the relative variances in spike counts?

Process 2 will have higher variance because the changing rate introduces additional variability beyond the Poisson variance.

If there are two possible stimuli stimulus1 and stimulus2 under maximum likelihood decoding one would decode a response by choosing stimulus 1 if proba(response|stimulus1) > proba(response|stimulus2)

True. With equal priors maximizing likelihood is equivalent to maximizing posterior probability.

The linear discriminant analysis (LDA) method of decoding used in class sometimes decodes the same stimuli as exact maximum likelihood decoding in special cases such as the response distributions being Gaussian with identical spherical shapes.

True. When covariance matrices are equal and spherical the LDA decision boundary matches the maximum likelihood boundary.

Consider a spike count produced by counting spikes from a Poisson process with rate r over a window of length T ms. If the rate r were to double the variance of the spike count would double.

True. For a Poisson process variance equals mean so doubling rate doubles variance.

Consider a spike count produced by counting spikes from a Poisson process with rate r over a window of length T ms. As the firing rate r increases the standard deviation of that spike count would increase proportionally to the square root of r.

Standard deviation = √(mean) = √(rT) so it increases as √r.

The ideal mathematical Poisson process has a Fano factor of 1.

True. Fano factor = variance/mean = 1 for a Poisson process.

Sample_list = [0 2]. What is the value of the sample standard deviation in this case?

The sample standard deviation is 1. Mean = 1 deviations -1 and +1 variance = 1 standard deviation = 1.

What statement best describes the role of inhibitory neurons in networks?

Inhibitory neurons tend to reduce the firing of downstream cells that they connect to by releasing inhibitory neurotransmitters.

What statement best describes the contributions of work by Hubel and Wiesel discussed in class?

Their work led to a hierarchical model of neural coding with complex cells downstream of simple cells in the sense that simple cells send inputs to complex cells.

In maximum likelihood decoding with two stimuli A and B having equal probability and Poisson spike counts with rates 50 vs 50.1 spikes/sec the error rate using maximum likelihood decoding is closer to 0.5.

With nearly identical means the distributions overlap almost completely making discrimination barely above chance level 0.5.

In the simplest analysis presented in class trajectories flowed downhill on the funnel function to values of firing rates that matched the smallest values of the weight matrix W_ij.

False. Trajectories flow to minima of the Lyapunov function which correspond to stored patterns not smallest weight values.

All other things being equal for HIGHER values of the constant potassium conductance gK extra added in class there was typically a suppressive effect on firing rates in which firing rates of the model neuron went down.

True. Higher potassium conductance increases outward current hyperpolarizing the cell and reducing firing.

When we are defining the decoding property of a cell which probability distribution are we defining?

We are defining P(stimulus | response) the probability of a stimulus given an observed neural response.

Consider decoding single trials as expressing the organism's point of view. If LDA for a single neuron gives fraction correct 0.1 for decoding direction of a drifting grating what additional considerations are relevant for predicting organism performance?

Both that LDA is approximate compared to optimal methods and that having many neurons improves performance as shown in homework.

In the firing rate model what does it mean that ri approaches f(inputsi) when inputs_i is constant?

The neuron reaches a steady state where its firing rate matches the f-I curve value for that input current with the time constant τ determining how quickly it approaches.

What is the difference between the firing rate model and the full Hodgkin-Huxley equations?

The firing rate model is a simplified phenomenological model where f captures the steady-state input-output relationship while Hodgkin-Huxley includes detailed ion channel dynamics.

Why is the RELU function considered linear in the context of the firing rate model?

RELU is linear above threshold (f(inputs)=inputs for inputs>0) making the equations linear in that regime while providing nonlinearity through the threshold.

In the linear recurrent network equation τ dr/dt = -r + Wr + stimulus why does an eigenvalue near 1 cause amplification?

The steady-state solution projects onto eigenvectors as cj = (s·vj)/(1-λj). As λj approaches 1 the denominator approaches zero causing amplification.

What happens if an eigenvalue has real part greater than 1 in the linear network?

The system becomes unstable with activity growing exponentially without bound rather than selectively amplifying the stimulus.

What is the difference between eigenvalues with real part near 1 versus eigenvalues with absolute value near 1?

Real part near 1 causes amplification in the direction of that eigenvector. Absolute value near 1 could come from complex eigenvalues with imaginary parts causing oscillations.

What are the three main ion currents in the Hodgkin-Huxley model?

Sodium current (INa) potassium current (IK) and leak current (I_L) each with their own conductances and reversal potentials.

What are the gating variables in the Hodgkin-Huxley model?

m is sodium activation h is sodium inactivation and n is potassium activation each with voltage-dependent rate equations.

How do voltage-dependent conductances differ from voltage-independent conductances in the Hodgkin-Huxley model?

Voltage-dependent conductances (gNa gK) change with membrane potential through gating variables while voltage-independent conductances (g_L) are constant.

What is the direction of potassium current when potassium channels open?

Potassium current flows OUT of the neuron (positive charge leaving) because E_K is typically around -77mV which is below resting potential.

In the Abbott model of synaptic depression what happens to P_rel when a spike occurs?

Prel is multiplied by the depression factor fD (typically less than 1) decreasing the release probability for subsequent spikes.

In the Abbott model of synaptic depression what happens to P_rel between spikes?

P_rel decays exponentially back to its baseline value P0 with time constant τ.

What is the steady-state relationship between P_rel and input rate r for a depressing synapse?

At steady state P_rel(r) × r ≈ constant for large r meaning the average transmission rate becomes independent of input rate.

Why does superposition hold for current-based RC circuits but not for conductance-based inputs?

Current-based inputs are linear in the differential equation while conductance-based inputs multiply with voltage creating nonlinear terms.

What is the Euler method formula for approximating differential equations?

x(t+Δt) = x(t) + Δt * f(x(t) t) starting from initial condition and iterating forward.

Why does the Euler method become more accurate with smaller timesteps?

Smaller timesteps better approximate the continuous derivative because the function is more linear over shorter intervals reducing truncation error.

What does conductance g represent in the RC circuit model of a neuron?

Conductance g = 1/R represents how easily ions flow through channels with higher conductance meaning more current for a given voltage difference.

What is the difference between conductance and resistance in neuronal modeling?

Conductance (g) is the inverse of resistance (R) with units of siemens. Higher conductance means less resistance to ion flow.

What does capacitance C represent in the RC circuit model of a neuron?

Capacitance represents the membrane's ability to store charge with charge Q = CV and capacitive current I_C = C dV/dt.

What physical property of neurons determines the capacitance C?

The lipid bilayer membrane acts as an insulator separating conductive fluids inside and outside creating a capacitor with capacitance proportional to membrane area.

How is the Fano factor calculated?

Fano factor = variance/mean. For a Poisson process this equals 1 regardless of the rate.

What does a Fano factor of 1 indicate about a spike train?

It indicates the spike train follows Poisson statistics where the variance equals the mean and spikes occur independently with constant rate.

Why does a Poisson process with constant rate still show trial-to-trial variability?

Because spikes are random events each trial is a different realization of the stochastic process with counts varying around the mean.

What does P(response | stimulus) represent in neural coding?

It represents the probability of observing a particular neural response given that a specific stimulus was presented capturing the variability of neural responses.

What does P(stimulus | response) represent in neural coding?

It represents the probability that a particular stimulus was presented given that a specific neural response was observed.

How do you obtain P(stimulus | response) from P(response | stimulus) using Bayes theorem?

P(stimulus|response) = P(response|stimulus)P(stimulus)/P(response) multiplying the likelihood by the prior and normalizing.

Why does a time-varying rate increase variance beyond a constant rate Poisson process?

The variance now includes both the Poisson variability at each instant and the variance due to rate changes across time.

What is the variance of a Poisson process with constant rate r over time T?

Var = rT equal to the mean. For time-varying rate the variance exceeds the mean.

If rate increases from 10 Hz to 20 Hz over the same time window how does the variance change?

Variance doubles from 10T to 20T maintaining variance = mean for constant rate Poisson.

What is the formula for standard deviation of a Poisson spike count?

Standard deviation = √(mean) = √(rT). So as rate increases standard deviation grows but slower than the mean.

How does the signal-to-noise ratio (mean/std) change with rate for a Poisson process?

SNR = mean/std = rT/√(rT) = √(rT) increasing with rate meaning faster firing gives relatively less noisy spike counts.

What does the close match between neurometric and psychometric functions in Newsome's work suggest?

That single neurons carry information sufficient to explain behavior and may be directly involved in decision-making.

What is the redundant codes hypothesis proposed by Zohary et al. 1994?

Neurons carry partially correlated signals so having many provides diminishing returns but ensures information survives noise and damage.

What assumptions underlie the Poisson process model of spike trains?

Spikes in different time bins are independent and the probability of a spike is proportional to bin width with constant instantaneous rate.

How do you generate a Poisson spike train in code?

For each time bin generate a uniform random number and place a spike if it is less than r×dt where r is rate and dt is bin width.

Derive the variance of a Poisson spike count from the bin method.

With bins of width dt each has mean r dt and variance r dt(1-r dt) ≈ r dt. Summing over T/dt bins gives variance = rT.

Why does SNR increase with firing rate even though variance also increases?

Mean increases linearly with rate while standard deviation increases only as √(rate) so the ratio grows as √(rate).

What does a higher SNR mean for neural coding?

Responses are more reliable relative to their variability making it easier to distinguish different stimuli.

In the Abbott model what is the depression factor f_D?

The multiplicative factor applied to Prel at each spike typically less than 1 causing depression. fD = 0.75 in many simulations.

In the Abbott model what is the recovery time constant τ?

The time constant for exponential return of P_rel to baseline between spikes typically hundreds of milliseconds.

How does the Abbott model achieve gain control?

By making Prel inversely proportional to r the product Prel × r becomes approximately constant so output rate is insensitive to input rate.

What advantage does gain control provide for a neuron receiving thousands of inputs?

It prevents saturation and maintains sensitivity to rate changes across all inputs regardless of their individual baseline rates.

What is the difference between synaptic depression and facilitation?

Depression decreases release probability with repeated spikes (low-pass filter). Facilitation increases release probability (high-pass filter burst detector).

What type of inputs do facilitating synapses preferentially transmit?

High-frequency bursts because each spike increases release probability making later spikes in the burst more effective.

In the context of the linear network what does the term trajectories flowed downhill on the funnel function refer to?

It refers to the Lyapunov function or energy function that decreases over time as the network settles into stable attractor states.

What is the Lyapunov function for Hopfield networks?

H(x) = -½ Σij Wij xi xj with dynamics minimizing this energy function.

What does it mean that trajectories do NOT flow to smallest weight values?

The energy minima correspond to stored patterns not to regions where weights are small which would actually be high energy states.

What is the difference between rate coding and temporal coding?

Rate coding uses spike count over time window. Temporal coding uses precise spike timing relative to stimulus or other spikes.

What is the difference between GABA-A and GABA-B receptors?

GABA-A fast ionotropic chloride channels. GABA-B slow metabotropic through G-proteins and potassium channels.

What is neuromodulation and how does it differ from classical transmission?

Neuromodulators alter neuronal properties over longer timescales rather than direct excitation/inhibition.

What are some examples of neuromodulators?

Dopamine serotonin acetylcholine norepinephrine each with specific receptors and functions.

What is the difference between feedforward and feedback inhibition?

Feedforward from different pathway. Feedback from recurrent collaterals of principal cells.

What is disinhibition and how does VIP interneuron circuitry achieve it?

VIP inhibits SST which inhibits pyramidal cells so VIP activation disinhibits pyramids.

What are gamma oscillations and what frequencies do they cover?

30-100 Hz associated with attention and memory often generated by PV interneuron-pyramidal cell loops.

What are theta oscillations and what frequencies do they cover?

4-8 Hz associated with navigation and REM sleep prominent in hippocampus.

What is the difference between local field potential (LFP) and spiking activity?

LFP reflects summed synaptic activity around electrode. Spiking reflects output of individual neurons near tip.

What is current source density (CSD) analysis?

Estimates transmembrane currents from spatial second derivative of LFP revealing sources and sinks across layers.

What is Bayes theorem in the context of decoding?

P(stimulus|response) = P(response|stimulus)P(stimulus)/P(response) allowing inversion from encoding to decoding.

What is the prior probability P(stimulus) in decoding?

The probability of each stimulus occurring before observing neural response often assumed equal.

What is the evidence P(response) in Bayes theorem?

The total probability of response across all stimuli acting as normalization constant.

What is maximum likelihood decoding?

Choosing stimulus that maximizes P(response|stimulus) equivalent to MAP when priors equal.

What is maximum a posteriori (MAP) decoding?

Choosing stimulus that maximizes P(stimulus|response) incorporating prior probabilities.

When would MAP and ML give different results?

When prior probabilities are not equal the MAP estimate is biased toward more likely stimuli.

What is the population vector decoding method?

Weight each neuron by its preferred stimulus and sum creating vector pointing toward decoded direction.

What is vector averaging in population decoding?

Population vector = Σ (responsei × preferredi) / Σ response_i giving direction estimate.

What is template matching in neural decoding?

Compare response vector to average response for each stimulus (templates) and choose closest match.