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Dynamical System in Neuroscience:

The Geometry of

Excitability and Bursting

پيمان گيفانی

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information processing by the brain.

DURING last few years we have witnessed a shift of the emphasisin the artificial neural network community toward

spiking neural networks .Motivated by biological discoveries,

many studies consider pulse-coupled neuralnetworks with spike-timing as an essential component in

In any study of network dynamics, there are two crucial issueswhich are

1) what model describes spiking dynamics of

each neuron and 2) how the neurons are connected.

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20 of the most prominent features of biological spiking neurons

20 of the most prominent features of biological spiking neurons. The goal of this section is to illustrate the richness and complexity of spiking behavior of individual neurons in response to simple pulses of dc current. What happens when only tens (let alone billions) of such neurons are coupled together is beyond our comprehension.

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Which Model to Use for Cortical Spiking Neurons?

To understand how the brain works, we need to combine experimental studies of animal and human nervous systems with numerical simulation of large-scale brain models.

As we develop such large-scale brain models consisting of spiking neurons, we must find compromises between two seemingly mutually exclusive requirements: The model for a single neuron must be:

1) computationally simple, yet 2) capable of producing rich firing patterns exhibited by real biological neurons.

Using biophysically accurate Hodgkin–Huxley-type models is computationally prohibitive, since we can simulate only a handful of neurons in real time. In contrast, using an integrate-and-fire model is computationally effective, but

the model is unrealistically simple and incapable of producing rich spiking and bursting dynamics exhibited by cortical neurons.

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Synaptic Dynamics

– Short-term Depression and Facilitation

– Synaptic Conductance

– Long-term Synaptic Plasticity

– Spike-timing in Neuronal Groups

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Rhythmic Activity

Rhythmic activity in the delta frequency range around 4 Hz. This is one of the four fundamental types of brain waves, sometimes called “deep sleep waves”, because it occurs during dreamless states of sleep, infancy, and in some brain disorders.As the synaptic connections evolve according to STDP, the delta oscillations disappear, and spiking activity of the neurons becomes more Poissonian and uncorrelated. After a while, gamma frequency rhythms in the range 30-70 Hz appear. This kind of oscillations, implicated in cognitive tasks in humans and other animals, play an important role in the activation of polychronous groups .

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Cognitive Computations

• Rate to Spike-Timing Conversion

Neurons in the model use spike-timing code to interact and form groups. However, the external input from sensory organs, such retinal cells, hair cells in cochlear, etc., arrives as the rate code, i.e., encoded into the mean firing frequency of spiking.

How can the network convert rates to precise spike timings?

Open circles - excitatory neurons, black circles - inhibitory neurons.

inhibitory postsynaptic potential (IPSP).

Notice that synchronized inhibitory activity occurs during gamma frequency oscillations. Thus, the network constantly converts rate code to spike-timing code (and back) via gamma rhythm. The functional implications of such a non-stop conversion are not clear.

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Representations ofMemories and Experience

hypothesize that polychronous groups could represent memories and experience. In the simulation above, no coherent external input to the system was present. As a result, random groups emerge; that is, the network generates random memories not related to any previous experience.

Persistent stimulation of the network with two spatio-temporal patterns result in emergence of polychronous groups that represent the patterns. the groups activate whenever the patterns are present.

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Consciousness When no stimulation is present, there is a spontaneous activation of

polychronous groups.

If the size of the network exceeds certain threshold, a random activation of a few groups corresponding to a previously seen stimulus may activate other groups corresponding to the same stimulus so that the total number of activated groups is comparable to the number of activated groups that occurs when the stimulus is present.

Not only such an event excludes all the other groups not related to the stimulus from being activated, but from the network point of view, it would be indistinguishable from the event when the stimulus is actually present. One can say that the network “thinks” about the stimulus. A sequence of spontaneous activations corresponding to one stimulus, then another, and so on, may be related to the

stream of thought and primary consciousness.

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NEURAL EXCITABILITY, SPIKING AND BURSTING

• The brain types of cells: neurons, neuroglia, and Schwann cells.

• neurons are believed to be the key elements in signal processing.

• neurons in the human brain each can have more than 10 000 synaptic connections with other neurons.

• Neurons are slow, unreliable analog units, yet working together they carry out highly sophisticated computations in cognition and control.

• Action potentials play a crucial role among the many mechanisms for communication between neurons.

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Synchronization and locking are ubiquitous in nature

In-phase SynchronizationAnti-phase Synchronization

                                                    

                                                    

Out-of-phase SynchronizationNo Locking

                                                  

  

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Neural Excitability • Excitability is the most fundamental property of neurons allowing communication via

action potentials or spikes.• From mathematical point of view a system is excitable when small perturbations near a

rest state can cause large excursions for the solution before it returns to the rest. • Systems are excitable because they are near bifurcations from rest to oscillatory

dynamics. • The type of bifurcation determines excitable properties and hence neuro-

computational features of the brain cells. Revealing these features is the most important goal of mathematical neuroscience.

• The neuron produse spikes periodically when there is a large amplitude limit cycle attractor, which may coexist with the quiescent state.

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• Most of the bifurcations discussed here can be illustrated using a two-dimensional (planar) system of the form

• Much insight into the behavior of such systems can be gained by considering their nullclines.

• the sets determined by the conditions f(x, y) = 0 or g(x, y) = 0. • When nullclines are called fast and slow, respectively. Since

the language of nullclines is universal in many areas of applied mathematics

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yxgy

yxfx

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An excitable system at an Andronov/Hopf bifurcation possesses an important information processing capability: Its response to a pair (or a sequence) of stimuli depends on the timing between the stimuli relative to the period of the small amplitude damped oscillation at the equilibrium.

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Selective communicationand multiplexing

The same doublet may or may not elicit response in a postsynaptic neuron depending on its eigenfrequency.This provides a powerful mechanism forselective communication between such neurons. In particular, such neurons can multiplex send many messages via a single transmission line.

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Phase of the sub-threshold oscillation

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Fast sub threshold oscillation

If a neuron exhibits fast subthreshold oscillation of its membrane potential, then its response to a brief strong input may depend on the amplitude and timing of the input.

If the input is weak, so that it never evokes an action potential, but can modulate the subthreshold oscillation, by changing its phase, so that the neuron would react dierently to a future strong pulse.

From the FM interaction theory it follows that the phase of subthreshold

oscillation can be affected only by those neurons with a certain resonant frequency.

By changing the frequency of the subthreshold limit cycle, the neuron can control the set of the presynaptic neurons that can modulate its dynamics.

The entire brain can rewire and regulate itself dynamically without changing the synaptic hardware.

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Bursters • When neuron activity alternates between a quiescent state and repetitive spiking,

the neuron activity is said to be bursting. It is usually caused by a slow voltage- or calcium-dependent process that can modulate fast spiking activity.

• There are two important bifurcations associated with bursting:  – Bifurcation of a quiescent state that leads to repetitive spiking. – Bifurcation of a spiking attractor that leads to quiescence.

• These bifurcations determine the type of burster and hence its neuro-computational features. 

An example of "fold/homoclinic" (square-wave) bursting. When a slow variable changes, the quiescent state

disappears via fold bifurcation and the periodic spiking attractor disappears via

saddle homoclinic orbit bifurcation

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Bursting• So far we have considered

spiking mechanisms assuming that all parameters of the neuron are fixed. From now on we drop this assumption and consider neural systems of the form

• Fast spiking

• Slow modulation

where u represents slowly changing parameters in the system.

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uxgu

uxfx

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Synchronization

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Type of Synchronization

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Coupling

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Oscillator & Traveling Wave • Understanding

synchronization properties of two coupled oscillators study dynamics of chains of n > 2 oscillators

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