1. Neural Oscillations Continued
Saee Paliwal
TNU

2. What is an oscillation? Simple Harmonic Oscillation
Mathematical foundation
amplitude
frequency
Limit Cycle of a neuron
phase
power ∝ amplitude
Neuronal oscillations: Van der Pol oscillators

3. Population oscillation
Coupled Oscillators
Local field potential of neuronal oscillations
This is the EEG signal

4. Coupled oscillations

5. An example of oscillations in vivo

6. Analyzing the EEG signal
Fourier transform:
decomposition of a signal from the time domain into the frequency domain
Four kinds of waves:
§beta (>13 Hz)
§alpha (8-13 Hz)
§theta (4-8 Hz)
§delta (0.5-4 Hz)
Frequency
Time
where do they come from?

7. Neural Mass Model: Jansen and Rit, David and Friston
Characteristics
Excitatory and Inhibitory connections
a few cubic mm of volume (~10^4-~10^7 neurons)
describe large populations of neurons with a handful of parameters (mean-field approximation)
Jansen and Rit model
Post-synaptic potential (PSP) model: convolution of impulse response and pre-synaptic input
Impulse response. H is the synaptic gain, t is the time constant
Membrane potential to rate for kth subpopulation.
c, r and e are population parametrs (e.g. volatge sensitivity)
The neural mass model is a description of the average activity of massively synchronous dendritic activity of pyramidal cells that generate M/EEG signals with a small number of state variables
These states summarise the behaviour of millions of interacting neurons. Basically, these models use two conversion operations (Jirsa and Haken, 1997; Robinson et al., 2001): a wave-to-pulse op- erator at the soma of neurons, which is generally a static sigmoid function, and a linear pulse-to-wave conversion implemented at a synaptic level, within the ensemble.
That symbol is a convolution. m is the pre-synaptic input (including pulse density) , h is the impulse response and v(t) is the PSP (post-synaptic membrane potential),
The excitatory (e) and inhibitory (i) kernels, he and hi, respectively, are parameterised by He,i and 􏰂e,i modelling specific properties of inhibition and excitation. The param- eters He,i tune the maximum amplitude of PSPs and 􏰂e,i are a lumped representation of the sum of the rate constants of passive membrane and other spatially distributed delays in the dendritic tree.
The transformation of average membrane potential of the population into an average action potential fired by the neurons is S. Membrane potential to rate.
Jansen and Rit, 1995

8. Circuit Diagram M/EEG signal
3 populations of neurons comprise a cortical area:
excitatory pyramidal cells
excitatory stellate cells
inhibitory interneurons
v is the membrane potential, m is the mean firing rate,
output signal
Pulse
The excitatory (e) and inhibitory (i) kernels, he and hi, respectively, are parameterised by He,i and 􏰂e,i modelling specific properties of inhibition and excitation. The param- eters He,i tune the maximum amplitude of PSPs and 􏰂e,i are a lumped representation of the sum of the rate constants of passive membrane and other spatially distributed delays in the dendritic tree.
David and Friston, 2003

9. Simulation of Oscillations
Jansen and Rit model with David and Friston extension

10. The power of synchrony
Long range synchrony: contributes to the “binding problem,” ie their integration represents a “gestalt” or patter/representation of a object
Visual Cortex: neural populations respond to color/shape/motion, and synchrony allows for the integration of these features into a percept (Eckhorn et al, 1988, Kreiter and Singer, 1996, Fries et al, 1997)
Feature bindng extends to oflactory system (Freeman et al, 1978) and the auditory system (Aersen et al, 1991)
It is also thought that synchronixation across several motor regions (ie motor cortex and spinal chord) lead to coordinated motor activity (van Wijk et al, 2012
Task-related Synchrony
binding across regions has been related to task-specific features
interocular rivalry, binding problem, coherent representations