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Dynamical System in Neuroscience: The Geometry of Excitability and Bursting

ینافيگ ناميپ

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

spiking neural networks.

Motivated by biological discoveries, many studies consider pulse-coupled neural networks with spike-timing as an essential component in

information processing

by the brain.

In any study of network dynamics, there are two crucial issues which 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

rhythm gamma frequency oscillations

to spike-timing code (and back)

via gamma

non-stop conversion are not clear.

. Thus, the network constantly converts rate code . The functional implications of such a

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Representations of Memories 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

polychronous groups.

is present, there is a

spontaneous activation

of If the

size of the network

exceeds

certain threshold

, a

random activation

of a few groups corresponding to a may activate other groups corresponding to the same stimulus so that the

total number

of activated groups is

previously seen stimulus 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

.

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.

10 11 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 Synchronization Anti-phase Synchronization Out-of-phase Synchronization No Locking

Neural Excitability

• • • • • Excitability is the

most fundamental action potentials or spikes.

property of

neurons

allowing

communication via

From

mathematical point of view

a system is

excitable

a rest state can cause large

excursions

when small

perturbations

for the solution before it returns to the rest. near

Systems are excitable

dynamics. because they are

near bifurcations

from

rest to oscillatory

The

type of bifurcation computational

determines

excitable properties

and hence

neuro-

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

x

' 

f

(

x

,

y

)

y

' 

g

(

x

,

y

) • • • 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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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 communication and multiplexing

The same doublet may or may not elicit response in a postsynaptic neuron depending on its

eigenfrequency

.

This provides a

powerful mechanism

for

selective 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

Fast sub threshold oscillation

29 If a neuron exhibits

fast subthreshold

its response to a the input.

brief strong input

oscillation of its membrane potential, then may depend on the

amplitude and timing

of 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.

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 30

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

x

' 

u

' 

f

(

x

,

u

) 

g

(

x

,

u

) where

u

represents slowly changing parameters in the system.

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Synchronization

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Type of Synchronization

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Coupling

47 • Oscillator & Traveling Wave Understanding synchronization properties of two coupled oscillators study dynamics of chains of

n >

2 oscillators

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