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4. Neurons in a network
Fundamentals of Computational Neuroscience, T. P. Trappenberg, 2002.
Lecture Notes on Brain and Computation
Byoung-Tak Zhang
Biointelligence Laboratory
School of Computer Science and Engineering
Graduate Programs in Cognitive Science, Brain Science and Bioinformatics
Brain-Mind-Behavior Concentration Program
Seoul National University
E-mail: [email protected]
This material is available online at http://bi.snu.ac.kr/
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Outline
4.1 Organizations of neural networks
4.2 Information transmission in networks
4.3 Population dynamics: modeling the
average behavior of neurons
4.4 The sigma node
4.5 Networks with non-classical synapses:
the sigma-pi node
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4.1 Organizations of neural networks

The high-order mental abilities
 An emerging property of specialized neural networks
 The number of neurons in the central nervous systems 1012
 We aim to understand the principal organization of
neuron-like elements and how such structures can support
and enable particular mental processes.

The anatomy of the brain areas
 Neocortex, cerebral cortex, cortex
 Cerebellum
 Subcortical area
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4.1.1 Neocortical organization

Brodmann’s cortical map
 52 cortical area

Functional correlates of different cortical areas
Fig. 4.1 Outline of the lateral view of the human brain including the cortex, cerebellum, and brainstem. The
neocortex is divided into for lobes. The numbers correspond to Brodmann’s classification of cortical areas.
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4.1.2 Staining techniques
Fig. 4.2 Examples of stained neocortical
slice showing the layered structure in
neocortex. (B) Illustration of different
staining techniques.
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4.1.3 Common neuronal types in the neocortex

Pyramidal cell

Stellate cell
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4.1.4 The layered structure of neocortex

A generally layered structure of the neocortex
 Laminar-specific structure
Fig 4.2 A
Fig 4.2 C
Fig. 4.2 Examples of stained neocortical slice showing the layered structure in neocortex. (A) Nissl stained visual cortex
showing cell bodies. (C) Different sizes of cortical layers in different areas.
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4.1.5 Columnar organization and cortical modules
Fig. 4.3 Columnar organization and topographic maps in neocortex. (A) Ocular dominance columns. (B)
Schematic illustration of the relation between orientation and ocular dominance columns. (C)
Topographic representation of the visual field in the primary visual cortex. (D) Topographic
representation of touch-sensitive areas of the body I the somatosensory cortex.
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4.1.6 Connectivity between neocortical layers
Fig. 4.4 Schematic connectivity patterns between neurons in a cortical layer. Open
cell bodies represent (spiny) excitatory neurons such as the pyramidal neuron and the
spiny stellate neuron. Their axons are plotted with solid lines that end at open
triangles that represent the axon terminal. The dendritic boutons are indicated by open
circles. Inhibitory (smooth) stellate neurons have solid cell bodies and synaptic
terminal, and the axons are represented by dashed lines.
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4.1.7 Cortical parameter
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4.2 Information transmission in networks
4.2.1 The simple chain

Biologically not reasonable
 A single presynaptic spike not sufficient to elicit a postsynaptic
spike.
 Synaptic transmission is lossy.
 The death of a single neuron would disrupt the transmission.
 Fig. 4.5 (A) A sequential transmission line of four nodes.
Parallel chains are made out of many such sequential
transmission lies without connections between them.
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4.2.2 Diverging-converging chains




The number of neurons, N
Divergence rate, m
Convergence rate, C
Fully connected network
N=m=C



Synaptic efficiency (weight), w
Feedback loop
Fig. 4.5 (B)
 Diverging/converging chains where each node can contact
several other nodes in neighboring transmission chains.
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4.2.3 Immunity of random networks to
spontaneous background activity (1)

Cortical neurons typically fire with some background
activity
 Mean 5Hz, variance 3Hz

Neuron that has 10000 excitatory dendritic synapses.
 Spiking arriving in each time interval (1ms)
 Mean μ = 10000 * 0.005 = 50
 Variance σ2 = 0.003

A spike arriving at a synapse with weight w=1 would
elicit a spike
 The neuron to be immune against the background firing.
w < 1/50 = 0.02
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4.2.3 Immunity of random networks to
spontaneous background activity (2)

To compare these value to experimental data
 How to measure the average synaptic efficiency
 To stimulate a presynaptic neuron while recording from the postsynaptic
neuron.
 Asynchronous gain
 The average number of extra spikes that are added to the spikes of a
postsynaptic neuron by each presynaptic spikes.
 If 100 presynaptic spikes (100Hz)
lead 2 postsynaptic spikes (2Hz)
during 1ms,
synaptic efficiency is 5/100 = 0.02
Fig. 4.6 Schematic illustration of the influence of a single
presynaptic spike on the average firing rate of the
postsynaptic neuron. The delay in the synaptic transmission
curve is caused by some delay, after which, on average, more
postsynaptic spikes are generated within a short time window
compared to the spontaneous activity of the neurons
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4.2.4 Noisy background

Large variability
 Ex) μ = 50, variance σ2 = 50





The probability of a postsynaptic spikes generated by the
background firing to be less than a certain value, pbg
The probability of having more than x simultaneous
presynaptic spikes
( y  )2
dy
 
1
2
Gaussian distribution
spikes
2
P( n
 x) 
e
(4.1)
If pbg = 0.1, x ≈ 59,
2  x
w < 1/59 ≈ 0.017
1
x
spikes
P(n
 x)  [1  erf (
)]
2
2 (4.2)
x    2erf -1 (1  2 p bg )
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(4.3)
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4.2.5 Information transmission in large
random networks



Previous condition, at least 59-50=9 additional presynaptic spikes to elicit
a meaningful postsynaptic spike
Large random networks
 1010 Neurons
 Each of these neurons connects to 10000 other neurons
 Stimulate 1000 neurons
 1000 * 10000 = 107 spikes transmitted
 The probability of a neuron receives
 A spike is 107 / 1010
 Two spikes is (107 / 1010)2=10-6
 Not sufficient to secondary spikes
A consequence of the small number of connections per neuron relative to
large number of neuron in the network
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4.2.6 The spread of activity in small
random networks

Netlets
 Small networks with only a small number of highly efficient
synapses.
 Only very few active presynaptic neurons can elicit a
postsynaptic spike in functionally correlated neurons.

Absolute Refractory Period
 Control the number of active neurons

Two Asymptotic model
 Small initial activity we get an inactive netlet
 Initial large activity we get a nearly maximally active netlet
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4.2.7 The expected number
of active neurons in netlets

The expected fraction of active nodes f
 f (t  1)  (1  f (t ))(1  e

 f (t )C
Fig. 4.7 The fraction of active nodes of netlets in the
time interval t + 1 as a function of the fraction of
active nodes in the previous time interval. The
different curves correspond to different numbers of
presynaptic spikes Θ that are necessary to elicit a
postsynaptic spike. (A) Netlets with only excitatory
neurons. (B) Netlets with the same amount of
excitatory and inhibitory connections.
1
( f (t )C ) n
)

(4.4)
n
!
n 0
the average number of synapses per neuron, C
the firing threshold, Θ
attractive fixpoint, firing rate 500Hz


4.2.8 Netlets with inhibition

Cortical neuron firing rate range
 10~100Hz

Inhibitory neurons
 the value still exceed.
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4.3 Population dynamics: modeling the
average behavior of neurons

Simulation of networks of spiking neurons
 Computing power problem
 Use average firing rate

Relationship of such models to population of spiking neurons
 What conditions these common approximations are useful and
faithful descriptions of neuronal characteristics


Rate models cannot incorporate all aspects of networks of
spiking neurons
Many investigations in computational neuroscience have used
such models
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4.3.1 Firing rate


The rectangular time window
The average temporal spike rate of a neuron with a window
size Δt: v(t )  number of spike in ΔT  1 t  ΔT / 2 (t 't f )dt '
ΔT

Gaussian window
v(t ) 
ΔT
1
2 




t  ΔT / 2
 (t 't f )e(t 't )
(4.5)
2
/ 2 2
dt '
(4.6)
Fig. 4.8 (A) Temporal average of a single spike
train with a time widow ΔT hat ahs to be large
compared to the average interspike interval
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4.3.2 Population averages

The average population activity A(t) of neurons
1 number of spikes in population of size N
T 0 T
N
1 t  T / 2 1 N
 lim
 (t 'tif )dt '


T 0 T t  T / 2 N
i 1
A(t )  lim

Very small time windows
A(t )dt 
1
N
(4.7)
N
  (t 't
i 1
f
i
)
(4.8)
Fig. 4.8 (B) Pool or local
population of neurons with
similar response characteristics.
The pool average is defined as
the average firing rate over the
neurons in the pool within a
relatively small time window.
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4.3.3 Population dynamics in response to
slowly varying inputs

The average behavior of a pool of neuron

dA(t )
  A(t )  g ( RI ext (t ))
dt
(4.9)
 Membrane time constant, τ
 Input current, I
 Activation function, g

Stationary state
dA
0
dt
A(t )  g ( RI ext (t ))
(4.10)
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4.3.4 Rapid response of population

Noise current at 100ms, increase external input current.

Population dynamics(eqn 4.9) vs average population spike rate
 response to rapidly varying input.

Adding noise, fluctuation, more realistic.
Fig 4.9
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Fig. 4.9 Simulation of a population of 1000
independent integrate-and-fire neurons with
a membrane time constant τm = 10 ms and
threshold θ = 10. Each neuron receives an
input with white noise around a mean RIext.
this mean is switched from RIext = 11 to RIext
= 16 at t = 100 ms. The spike count of the
population almost instantaneously follows
this jump in the input, whereas the average
population rate, calculated from in eqn 4.9
with a linear activation function, follows this
change input only slowly when the time
constant is set to τ = τm .
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4.3.5 Advanced descriptions of population
dynamics



ui (t )   (t  t f )   wij (t  t jf )
j t
Spike response model
t
(4.11)
Ignored ε term, i for the postsynaptic neurons, average
0


ij
synaptic efficiency
N (4.12)
No spike-time adaptation in neuron. the mean influence of the
postsynaptic potential
f
f
j

u (t )  0   (t ' ) A(t  t ' )dt '
0


(4.13)
Noise A(t )   Pu (t | t ) A(t )dt (4.14)
g ( x) 

In slowly varying input, eqn 4.9 with a gain function, t
t
f
f
(4.15)
f
1
ref
  log( 1 
Fig. 4. 10 (A) The gain function of eqn 4.15 that
can be used to approximate the dynamics of a
population response to slowly varying inputs
(adiabatic limit). (B) Examples of physiological
gain functions from a hippocampal pyramidal
cell. The discharge frequency is based on the
inverse of the first interspike interval after the
cell started to respond to rectangular current
pulses with different strength
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1
)
x
4.4 The sigma node



McCulloch-Pitts node
Rate model
The timing of spikes is irrelevant.
 But spike times play critical role for fast response
4.4.1 Minimal neuron model


Sigma node
Rate value of neuronal groups, r
Fig. 4.11 Schematic summary of the
functionality of a sigma node most
commonly used in networks of
artificial neurons. Such a node
weights the input value of each
channel with the corresponding
weight value of that channel and
sums up all these weighted inputs.
The output of the node is then a
function of this internal activation.
hi  wi riin
h   hi
i
(4.16)
(4.17)
r out  g (h)
(4.18)
r out  g ( wi riin )
i
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(4.19)
4.4.2 Common activation functions

Activation function, g

generalize the sigmoid function
g
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sig
1

1  exp(   ( x  x0 ))
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(4.20)
4.4.3 The dynamic sigma node

The discrete sigma node to continuous dynamics
h(t  t )  (1 

t

)h(t ) 
t

 r
i i
i
(4.21)
Continuous, limΔt→0.

h(t  t )  h(t )
dh(t )
 h(t )   i ri

 h(t )   i ri
(4.22)
t
dt
i
i
Fig. 4.12 Time course of the activation of
a leaky integrator node with initial value h
= 0. In the lower curve no input Iin was
applied leading to an exponential decay.
The upper curve corresponds to a constant
input current Iin = 0.5. The resting
(4.23) activation of the node was set to hrest = 0.1.
4.4.4 Leaky integrator characteristics

The leaky integrator dynamics.
  dh(t )  h(t )   r without external input

dt
i i
i
t
 Solve h(t )  h(0)e   h rest (4.24)
in
i ri change behavior
 I 
i
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4.4.5 Discrete formulation of continuous
dynamics

The exponential response to short inputs
h(t  t )  (1  e

t

t
)h(t )  e 
 r
i i
i


(4.25)
Taking time steps on a logarithmic scale
Method
 Euler method
 Higher-order Runge-Kutta method
 Adaptive time step algorithm
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4.5 Networks with non-classical synapses:
the sigma-pi node

Sigma node
 A very rough abstraction of real neuron

Interaction of different ion channels
 Information-processing
 Nonlinear interaction between input channels
 Average firing rate
4.5.1 Logical AND and sigma-pi nodes

Nonlinear interaction between two ion channels
 Firing threshold
 Requires at least two spikes in some temporal proximity
 Correspond to a logical AND function
in in
h

w
r
 Generalize this idea for model, sigma-pi node i  jk j rk
jk
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(4.26)
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4.5.2 Divisive inhibition

Interaction between an excitatory synapse and an
inhibitory synapse
 Divisive inhibition
Shunting inhibition
hi   wik riexcitatory / rkinhibitory
jk
(4.27)
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4.5.3 further sources of modulatory effects
between synaptic inputs
Fig. 4.13 Some sources of nonlinear (modulatory) effects between synapses as modeled by sigma-pi nodes. (A) shunting (divisive) inhibition,
which is often recorded as the effect of inhibitory synapses on the cell body. (B) The effect of simultaneously activated voltage-gated excitatory
synapses that are in close physical proximity to each other (synaptic clusters) can be larger than the sum of the effect of each individual synapse.
Examples are clusters of AMPA ad NMDA type synapses. (C) some cortical synaptic terminals have nicotinic acetylcholine (ACh) receptors.
An ACh releases of cholinergic afferents can thus produce a larger efflux of neurotransmitter and thereby increases EPSPs in the postsynaptic
neuron of this synaptic terminal. (D) Metabotropic receptors can trigger intracellular messengers that can influence the gain of ion channels. (E)
Ion channels can be linked to the underlying cytoskeleton with adapter proteins and can thus influence other ion channels through this link.0
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Conclusion


The brain does display characteristic neuronal organizations
Properties of networks of spiking neurons
 The spread of neuronal activities through random networks
 Transmission of information
 A sensible activity in random recurrent networks
 The self-organization of synaptic efficiencies

The sigma node
 Modeling the average firing rate of populations of neurons
 Sigma-pi node
 Rate model
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