Why are cortical spike trains irregular?

1. How Why are cortical spike trains irregular? Arun P Sripati & Kenneth O Johnson Johns Hopkins University

2. Background/Motivation • The real reason: (What are the changes in firing variability during a shift in attention, and how are they caused?) • The spiking mechanism is known to be precise, so firing variability must be due to irregularly arriving synaptic inputs (Calvin & Stevens, 1968) • Irregular firing can be modeled as a random walk to threshold (Gerstein and Mandelbrot, 1964, etc, etc.) • Recently: increased emphasis on cortical neurons, attempts to replicate high Inter Spike Interval (ISI) variability in realistic neuronal models. (ISI CV = std/mean of ISI)

3. Outline • Take a biophysically realistic model that is widely used to account for the irregular activity of cortical neurons (the conductance-based integrate and fire model, driven by balanced excitation and inhibition). • Reduce it to obtain a random walk (i.e. the random walk parameters are directly obtained from the conductance-model parameters) • Obtain analytical expressions for the mean and the variance of the random walk, and check with the conductance-based model. • Compare with current injections in vitro (the real test)

4. A model that explains high ISI CV • Cortical neurons are driven by ~1000s inputs, 80% of which are excitatory. All inputs are assumed to be Poisson processes. • If all inputs were equally strong, the dominance of excitation would cause a regular progression to threshold. BUT: real spike trains are far more irregular than that. (simulations of a conductance based model)

5. The conductance-based model: details Model has 160 Exc, 40 Inh inputs. Essentially: A balanced conductance-based model accounts for the Poisson-like firing of cortical neurons.

6. Simplifying the conductance-based model Spiking variability arises from stochastic fluctuations in the membrane potential, which implies a random walk. Since the model neuron is driven by a large number of inputs, the distribution of changes in the sub-threshold membrane potential over short intervals must be Gaussian. Thus the random walk has Gaussian step increments. After a spike, the membrane potential quickly reaches a steady state about which it subsequently fluctuates. This is because there is virtually no inhibition at the reset (due to the inhibitory reversal potential being close to reset). Thus the random walk effectively begins at this “steady state” membrane potential. For the same reasons, the membrane potential rarely goes close to the reset potential  the random walk is constrained from below.

7. The random walk model Thus, the random walk has: • An absorbing barrier (Threshold) • An elastic reflecting barrier (to prevent negative excursions) • A reset value • Gaussian distributed step increments • A discrete time step, chosen to reflect the characteristic correlation of membrane potential (~1ms). Threshold Steady state Reflecting barrier reset

8. Predicting the input-output relationship

9. Predicting the ISI CV

10. Predicting the ISI Histogram • ISIH is not exponential (is probably Inverse Gaussian…)

11. Testing our predictions in vitro (ongoing): • Inject colored noise (White gaussian noise, filtered with an RC filter) into a neuron in current clamp mode, and observe the spiking statistics as a function of the input current statistics.