The equations for a model describing oscillation in magnetic field polarity are:
ID: 3372768 • Letter: T
Question
The equations for a model describing oscillation in magnetic field polarity are:>
x' = -ux+yz
y' = -uy + (z-a)z
z' = 1-xy
where a and u are positive constants. Find the equilibria for this system for a = u = 1, and compute the eigenvalues of the linearized system at those equilibria.>style="font-size: 11.000000pt; font-family: 'cmmi10'">style="font-size: 11.000000pt; font-family: 'cmr10'">style="font-size: 11.000000pt; font-family: 'cmmi10'">style="font-size: 11.000000pt; font-family: 'cmr10'">style="font-size: 11.000000pt; font-family: 'cmmi10'">style="font-size: 11.000000pt; font-family: 'cmr10'">style="font-size: 11.000000pt; font-family: 'cmmi10'">style="font-size: 11.000000pt; font-family: 'cmr10'">
Explanation / Answer
Neural oscillation is rhythmic or repetitive neural activity in the central nervous system. Neural tissue can generate oscillatory activity in many ways, driven either by mechanisms localized within individual neurons or by interactions between neurons. In individual neurons, oscillations can appear either as oscillations in membrane potential or as rhythmic patterns of action potentials, which then produce oscillatory activation of post-synaptic neurons. At the level of neural ensembles, synchronized activity of large numbers of neurons can give rise to macroscopic oscillations, which can be observed in the electroencephalogram (EEG). Oscillatory activity in groups of neurons generally arises from feedback connections between the neurons that result in the synchronization of their firing patterns. The interaction between neurons can give rise to oscillations at a different frequency than the firing frequency of individual neurons. A well-known example of macroscopic neural oscillations is alpha activity.
Neural oscillations were observed by researchers as early as Hans Berger, but their functional role is still not fully understood. The possible roles of neural oscillations include feature binding, information transfer mechanisms and the generation of rhythmic motor output. Over the last decades more insight has been gained, especially with advances in brain imaging. A major area of research in neuroscience involves determining how oscillations are generated and what their roles are. Oscillatory activity in the brain is widely observed at different levels of observation and is thought to play a key role in processing neural information. Numerous experimental studies indeed support a functional role of neural oscillations; a unified interpretation, however, is still lacking.
Simulation of neural oscillations at 10 Hz. Upper panel shows spiking of individual neurons (with each dot representing an individual action potential within the population of neurons), and the lower panel the local field potential reflecting their summed activity. Figure illustrates how synchronized patterns of action potentials may result in macroscopic oscillations that can be measured outside the scalp.
Contents [hide]
1 Overview
2 Physiology
2.1 Microscopic
2.2 Mesoscopic
2.3 Macroscopic
3 Mechanisms
3.1 Neuronal properties
3.2 Network properties
3.3 Neuromodulation
4 Mathematical description
4.1 Single neuron model
4.2 Spiking model
4.3 Neural mass model
4.4 Kuramoto model
5 Activity patterns
5.1 Ongoing activity
5.2 Frequency response
5.3 Amplitude response
5.4 Phase resetting
5.5 Additive response
6 Function
6.1 Pacemaker
6.2 Central pattern generator
6.3 Information processing
6.4 Perception
6.5 Motor coordination
6.6 Memory
6.7 Sleep and Consciousness
7 Pathology
7.1 Tremor
7.2 Epilepsy
8 Applications
8.1 Brain-computer interface
9 Examples
10 See also
11 References
12 Further reading
13 External links
Overview[edit source | editbeta]
Neural oscillations are observed throughout the central nervous system and at all levels, e.g., spike trains, local field potentials and large-scale oscillations which can be measured by electroencephalography. In general, oscillations can be characterized by their frequency, amplitude and phase. These signal properties can be extracted from neural recordings using time-frequency analysis. In large-scale oscillations, amplitude changes are considered to result from changes in synchronization within a neural ensemble, also referred to as local synchronization. In addition to local synchronization, oscillatory activity of distant neural structures (single neurons or neural ensembles) can synchronize. Neural oscillations and synchronization have been linked to many cognitive functions such as information transfer, perception, motor control and memory.[1][2][3]
Neural oscillations have been most widely studied in neural activity generated by large groups of neurons. Large-scale activity can be measured by techniques such as electroencephalography (EEG). In general, EEG signals have a broad spectral content similar to pink noise, but also reveal oscillatory activity in specific frequency bands. The first discovered and best-known frequency band is alpha activity (8
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