Literature
Theoretical Neuroscience
Computational and Mathematical Modeling of Neural Systems
Peter Dayan and L. F. Abbott
MIT Press 2001/2005
Principles of Neural Science
Kandel ER, Schwartz JH, Jessell TM
4th ed. McGraw-Hill, New York. 2000
Spiking Neuron Models
Single Neurons, Populations, Plasticity
Wulfram Gerstner and Werner M. Kistler
Cambridge University Press (August 2002)
http://icwww.epfl.ch/~gerstner/SPNM/SPNM.html
Links
Munich Bernstein Center for Computational Neuroscience
http://www.bccn-munich.de/
Scholarpedia (Computational Neuroscience)
http://www.scholarpedia.org/article/Encyclopedia_of_Computational_Neuroscience
GEneral NEural SImulation System
http://www.genesis-sim.org/GENESIS/
NEURON for empirically-based simulations of
neurons and networks of neurons
http://neuron.duke.edu/
http://www.neuron.yale.edu/
Single compartment models: membrane
Dayan & Abbott 2000
V : membrane potential
Ie : external current
Rm : membrane resistance
Q : charge
V : membrane potential
Cm : membrane capacitance
Neuronal surface: 0.01-0.1 mm2
Approx. 109 singly charged ions necessary for resting potential –70 mV.
Single compartment models
Dayan & Abbott 2000
Ek : reversal potential
gk : conductance
C : membrane capacitance
Equivalent circuit of a single compartment neuron model.
Synaptic conductance (s) depends on activity of a pre-synaptic neuron.
Voltage-dependent conductance (v) depend on membrane potential V.
Hodgkin-Huxley model
Ek : reversal potential
gk : conductance
m, n, h : gating variables
Hodgkin-Huxley: system of four non-linear differential equations
Hodgkin-Huxley model
Potassium current: gK n4 (u(t)-EK)
Gating variable n describes the
process of transition of an “activation
particle” from the open to the closed
state and vice versa.
Probability of being open: n
Probability of being closed: 1-n
For a potassium channel (delayedrectifier K+) to open, 4 activation
particles have to open ! n4.
" and # are voltage-dependent rate
constants and describe how many
transitions can occur between both
states.
Hodgkin-Huxley model
differential equations of the gating variables can be rewritten as voltage
dependent low-pass filters
different time scales: time constant of m never exceeds 1ms, other
variables show slower time constants of roughly equal magnitude
gating variables vary between 0 and +1, n0(u)$1-h0(u)
Hodgkin-Huxley model
generation of an action potential:
m opens the sodium channel (fast,
transient conductance)
the membrane potential rises
both h and m are non-zero ! Na+ influx
this causes the spike
then h closes the sodium channel (slow)
n opens the potassium channel (slow,
persistent conductance)
the membrane potential decreases
=> combination of Na and K currents
causes the action potential
Some features of the Hodgkin-Huxley model
Refractory period: if the interval between current pulses is too short, a
new action potential cannot be generated.
Rebound firing: a hyperpolarizing
current causes a “rebound spike”.
Activation function: steady-state
firing rate as a function of somatic
input current
Hodgkin-Huxley model
Gerstner & Kistler 2002
Response of the Hodgkin-Huxley model to step current input.
Three regimes:
I: no action potential is initiated (inactive regime).
S: a single spike is initiated by the current step (single spike regime).
R: periodic spike trains are triggered by the current step (repetitive firing).
Stability: fixed points and limit cycles
Wanted: stable and unstable regions depending on parameters
1)! Find a fixed point
2)! Linearize
3)! Determine whether it is stable
Useful: phase plane analysis
BUT: only for two variables (some theorems do not hold for higher dimensions)
Try to simplify model to just two differential equations
u˙ = ( F(u,w) + RI ) /"
w˙ = G(u,w) /" w
To check for stability, compute eigenvalues of the linearized system.
Eigenvalues and Eigenvectors
An eigenvector of a given linear transformation is a vector whose direction is not
changed by that transformation. The corresponding eigenvalue is the proportion by
which an eigenvector's magnitude is changed.
Ax = "x
( A # "E ) x = 0
The eigenvectors x of the matrix A fulfil these equations, with % being the
corresponding eigenvalue. E is the unit matrix.
The eigenvalues are computed by solving for the determinant.
det(A " #E) = 0
FitzHugh Nagumo model: Phase plane analysis
Simpler model: FitzHugh-Nagumo
1 3
u˙ = u " u " w + I
3
w˙ = # $ (b0 + b1u " w)
Phase plane analysis
Linearization of the system at the
intersection (fixed point) by Taylor
expansion to first order.
Solving for the eigenvalues: both
eigenvalues must have real parts smaller
than zero.
=> Necessary and sufficient condition for
stability of the fixed point.
trace
determinant
Phase plane analysis
Parameters: b0=2, b1=1.5, &=0.1
Fixed point for I=0 at u=-1.54, w=-0.32
For this fixed point, the stability condition
is fulfilled.
Stable solution for
Phase plane analysis
I=2: instable fixed point, limit cycle
Phase plane analysis
http://www.scholarpedia.org/article/FitzHugh-Nagumo_Model
Different regimes separated by the null-clines
Phase plane analysis
Many of the properties of the Hodgkin-Huxley model are easily explained by
the FitzHugh-Nagumo model
“Excitation Block” in the FitzHughNagumo model
The mechanism of rebound firing explained
by phase plane analysis
http://www.scholarpedia.org/article/FitzHugh-Nagumo_Model
FitzHugh-Nagumo vs. Hodgkin-Huxley
Similarity of a reduced version of the Hodgkin-Huxley model
and the FitzHugh-Nagumo model
Reduced Hodgkin-Huxley
FitzHugh-Nagumo
Numerical simulation in MATLAB
1)!
Script-based numerical simulations
Uses MATLAB programming language
Various solvers for ordinary differential equations available
Sometimes much more flexible than Simulink
Requires some programming skills
2)!
Simulink-based numerical simulations
Simulink graphical user interface
Very convenient for rapid development
Provides inbuilt visualization (scopes)
Subsystems for modular architectures
Vectorization for simulation of large numbers of neurons
Numerical simulation
du
"
= E + R# I $ u
dt
Basic differential equation.
d u u( t ) # u( t # $t )
"
$t
dt
Express the derivative by a difference.
ui := u(t i ) t i+1 := t i + "t
u˙ i+1 = ( E + R" Ii # ui ) /$
"d u%
ui+1 ) ui
u˙ i+1 = $ ' (
# d t & i+1
*t
+ ui+1 = ui + u˙ i+1 , *t
For convenience, use indices.
Calculate derivative dui+1/dt at time ti+1 from
values at time ti.
Integration step: calculate ui+1 from the
known value ui and the derivative.
(Euler one-step method)
Note: for good convergence, 't has to be
much smaller than (.
Levels of Abstraction
Level I-III
Uses Hodgkin-Huxley style membrane equations or even
more detailed equations for ion channel dynamics and
synaptic transmission.
Level IV
Linear-nonlinear cascades, phenomenological model,
easier to fit to experimental data.
Level V
Only conditional probability of a response R depending on
a stimulus S is considered.
Herz et al., Science 314, 2006
Leaky integrate and fire model
Membrane potential between spikes
follows a linear differential equation
(lowpass filter, time constant (=RC).
At threshold ), the neuron fires a spike
(usually modelled as Dirac pulse), and the
membrane potential is reset to ur.
Thus, the time course of the membrane
potential in response to a constant current
I0 can be calculated analytically by
integrating the differential equation.
The solution is:
Leaky integrate and fire model
To calculate the firing rate of the neuron in
response to a constant input current, we
simply set u(t)= ), and calculate the time
between two successive spikes. The firing
rate is then f=1/!t.
In many applications, an absolute
refractory time tabs is introduced to account
for the fact that two spikes cannot follow
each other immediately.
Spike rate adaptation
Many real neurons show spike rate adaptation, i.e., the firing rate in response to a
constant current decreases over time ((=100ms). The original integrate and fire
neuron can be modified to account for this behaviour by adding an additional current
with a time-dependent conductance grsa (modelled as K+ conductance).
When the neuron fires, the conductance is increased by a constant amount.
Refractory period
Instead of introducing an artificial absolute refractory period, a more realistic relative
refractory period can be simulated by the same mechanism as spike rate adaptation,
but with a much shorter time constant ((=2ms) and larger increase of the leak
conductance following an action potential.
Synaptic transmission
Information is collected via the dendritic tree.
Increase in membrane potential leads to an action
potential at the axon hillock.
The action potential is actively moving along the axon.
It arrives at the synapses, causes release of
neurotransmitters.
Neurotransmitters open ion channnels at the
postsynaptic neuron.
This leads to a change in conductance, and
consequently to a change in the membrane potential of
the postsynaptic neuron.
Synaptic transmission: mechanism
Action potential causes Ca2+ channels to open.
Calcium influx causes vesicles to fuse with the membrane and release transmitter.
Transmitter causes receptor channels to open, an excitatory postsynaptic potential
(EPSP) can be observed .
Synaptic transmission: transmitters and receptors
Excitatory
Transmitter: glutamate
Results in opening of Na+ and Ka+ channels at the postsynaptic membrane
Reversal potential is at 0 mV (i.e., at 0 mV, no EPSP is visible)
Ionotropic glutamate receptors: AMPA, kainate, NMDA
AMPA: "-amino-3-hydroxy-5-methylisoxazole-4-propionic acid
NMDA: N-methyl-D-aspartate
AMPA
short time constant (2 ms)
NMDA
long time constants (rise: 2 ms, decay 100 ms)
voltage dependent: Mg2+ blocks channel at –65 mV and opens upon depolarization
needs extracellular glycine as cofactor
Metabotropic glutamate receptors: indirectly gate channels through second messengers
Synaptic transmission: transmitters and receptors
Inhibitory
Transmitter: GABA and glycine
Results in opening of Cl- channels at the postsynaptic membrane
Reversal potential is at -70 mV (i.e., at -70 mV, no IPSP is visible)
GABA-receptors: GABAA (ionotropic) and GABAB (metabotropic)
GABAA
longer time constant 10 ms, Cl- conductance
GABAB
Opening of K+ channels, reversal potential –80 mV
Single channel conductance of glycine is larger (46 pS) than that of GABA (30 pS)
Basic equation for synaptic input
du
Cm
= gl " ( E l # u) + gs " w s " ps " ( E s # u)
dt
ps : fraction of open channels
d ps
ps
= # + &% ( t # t k )
ws : synaptic weight
dt
$s k
E : synaptic resting potential
s
!s : synaptic time constant
Basic differential equation of the leaky integrate and fire model
with synaptic input.
The fraction of open channels determines the synaptic conductivity.
The product of conductivity and difference between membrane potential and
synaptic resting potential determines the synaptic current.
Synaptic transmission: change in conductance
Synapse as change in input current: the effective afferent current I(t) of a neuron
depends on the input from N synapses with weights wi. The kth spike arrives at time tik at
synapse i.
More realistic: change in conductance; involves not only the arriving spikes, but also the
membrane potential of the postsynaptic neuron.
If the reversal potential of the synapse uE is sufficiently high (true for excitatory
synapses, e.g. uE=0), then the dependence of the postsynaptic current on the membrane
potential can be neglected.
The effect of inhibitory spikes depends on the
membrane potential of the postsynaptic neuron.
For inhibitory synapses, this is not the
case: uE is usually equal to the reversal
potential of the neuron itself.
Synaptic transmission: modelling
Synaptic transmission can be modelled by lowpass filter dynamics.
The dynamics of the EPSP can be described by one or multiple time constants.
Synaptic transmission
Train of input spikes
(from one or more neurons, s is
linear!)
Change in conductance s (blue)
Change in input current (red)
Membrane potential
Output spikes
Synaptic transmission: shunting inhibition
excitation only
“silent” inhibition
shunting inhibition
Strong inhibitory inputs do not need to generate an IPSP to affect output.
Inhibition changes the conductance of the membrane: the neuron becomes leakier!
Spike timing
Spike timing makes a difference!
Left:
unsynchronized input spikes (i.e., spikes do not occur at the same time) from 3
neurons do not cause the postsynaptic cell to fire
Right: the same number of input spikes, again from 3 neurons, lead to firing of the
postsynaptic cell, if spike times are synchronized
From real spikes to spike events
Get spikes from measured data:
Determine a threshold for spike
discrimination
Detect maxima above threshold
Distribution of inter-spike intervals
Probability distribution of time
between action potentials (interspike interval, ISI) shows two peaks.
This indicates that the neuron
responded to some external events.
Another way to look at it: scatter plot
of successive ISIs.
How can we describe spiking in
terms of statistics?
Probabilistic spike generation
Independent spike hypothesis:
The generation of each spike is a random process driven by an underlying
continuous signal r(t) and independent of all other spikes. The resulting spike train
is thus a Poisson process.
The neural response function !(t) is the sum of spikes with each spike being
modelled as Dirac pulse "(t).
k
"(t) = %# (t $ t i )
i=1
The spike count can thus be expressed as integral over time:
n=
The firing rate is the expectation of the neural response function:
t2
r(t)dt
The average spike count is then n =
t1
For small time intervals, the expected spike count thus becomes
"
#
t2
t1
" (t)dt
r(t) = "(t)
n = r(t)" #t
If !t is small enough, the probability of more than one spike occurring can be
ignored and thus
P (1 spike in [ t - "t /2,t + "t /2[) = r(t)# "t
Poisson spikes
Probability that a neuron discharges n spikes in the
time interval !t can be described by the Poisson
distribution (r: mean firing rate):
(r# "t) n $r#"t
#e
P("t,n) =
n!
Probability that the next spike occurs before the time
interval T has passed:
P(t < T) = 1 " P(T,0) = 1 " e "r#T
Hence, the probability to find an ISI of T is:
P(T) =
d
dT
P(t < T) = r" e #r"T
The mean duration between events is thus
The mean spike count is
This is the exponential distribution.
T =
$
#
0
T" P(T)dT = 1/r
n = r" #t and the variance is "n2 = r# $t
Thus, for Poisson spike trains, the Fano factor is
"n2
=1
F=
n
Poisson spikes
Example of simulated Poisson spike trains
Interspike intervals are exponentially distributed
Distribution of inter-spike intervals
An alternative to the exponential distribution
(see, e.g., Shadlen & Newsome 1998).
is the Gamma distribution (assume that the
neuron discharges only after k events,
or, in other words, only every k'th time):
T k "1 k "r$T
P(T) =
$r $e
#(k)
k can thus be conceived as neuronal threshold.
Empirical probability distribution of ISIs of a
real neuron fitted with the gamma distribution.
Reliability of Spiking Neurons
Ermentrout et al. 2008, after Mainen et al. 1995
Noisy input as trigger for reliable spiking
Neurons are sensitive to changes in input rather than input per se.
From spikes to firing rate
Compute the instantaneous firing rate
from spike events
ti : time of i th spike
f: firing rate
The instantaneous firing rate is
assumed to be constant between
spikes.
Technically, however, it does not
exist between spikes.
From spikes to firing rate
Convolution (Faltung):
Dayan & Abbott 2000
Different methods to approximate firing rate. A: spike train, B: discrete rate,
bins of 100ms, C: sliding window (100ms), D: Gaussian window ("=100ms),
E: alpha function window (1/#=100ms)
From spikes to firing rate
Top: Firing rate computed via interspike interval or convolution with a Gaussian window
(corresponding to a probability distribution of each spike).
Bottom: synaptic current computed from spike train or from firing rate.
From spiking neurons to firing rate
The current due to synaptic input is
approximately described by leaky integration of
the incoming spikes.
The average firing rate r can be expressed as
convolution (filtering) of the spike train with a
window function (kernel) W(t), or as sum over
the window functions at the times where spikes
occur.
The sum over the incoming spikes can thus be
replaced by the respective firing rate, if the
interspike intervals are short compared to the
synaptic time constant.
The firing rate of the postsynaptic neuron is
determined by the neuron’s response function
F(I(t)), which can be approximated by a
threshold-linear function for the ideal integrateand-fire neuron.
From spiking neurons to firing rate
Firing rate networks come in various flavours.
Usually, the dynamics are modelled by a single
time constant, which may vary among neuron
populations or for different types of synapses.
The time constant describes either the dynamics
of synaptic transmission, or that of the neuron
itself. In the latter case, the time constant will
usually will smaller than the membrane time
constant.
A combination of both is possible, of course.
From spiking neurons to firing rate
N1
NMDA synapse
N2
Injected current I(t)
For long synaptic time constants, the spike rate response of the postsynaptic neuron is
determined by the synaptic time constant
Feedforward network
input units
synaptic weights
output unit
The simplest case of a network is a feedforward
network with n input units and a single output
unit.
ri(t) : firing rate of input unit i
% n
(
˙I ( t ) = ' # w " r ( t ) $ I ( t )* / + = ( w ! r( t ) $ I ( t )) / +
i
i
& i=1
)
I(t) : synaptic current of the output unit
y ( t ) = [ I ( t )]
wj
+
y(t) : firing rate of the output unit
: synaptic weight of the ith input
[]+ : linear-threshold nonlinearity
The synaptic weighting of the n inputs can be written as vector dot product (scalar product).
a
b
In Euclidean geometry, the dot product can be
visualized as orthogonal projection:
If b is a unit vector, then the dot product of a
and b is the orthogonal projection of of a on b.
For our simple network, the steady state is
reached when
Feedforward network
input units
synaptic weights
For our simple network, the output will
become maximal, if the input vector and the
weight vector are collinear.
output unit
In the steady state, this network is very similar to the so-called perceptron.
The main difference is the non-linearity, which is a ! function for the perceptron (i.e., the
output is either 0 or 1).
The perceptron is a binary linear classifier, i.e. it
responds with 1 if the input vector is on one side of the
decision boundary (a hyperplane), and with 0 if it is on
the other.
Simple two-dimensional
case with equal weights
Boundary at
Feedforward network
input units
synaptic weights
output unit
More than one output unit: the weights can be
written as a matrix equation.
First, recall that the dot product can be
rewritten as matrix multiplication:
Multiple outputs can be easily expressed by matrix multiplication:
To understand the operation performed by the network, we consider the simple twodimensional case.
The weights have been chosen so that the matrix W
implements a rotation of the input space (x: 2D input; y:
2D output).
In the general case (n inputs, m outputs), the network
performs a transformation and/or projection of the ndimensional input vector onto an m-dimensional output
vector.
Attractor networks
An attractor network is a network of nodes (i.e.,
neurons in a biological network), often
recurrently connected, whose time dynamics
settle to a stable pattern.
Chris Eliasmith (2007), Scholarpedia, 2(10):1380.
N nodes connected such that their global dynamics becomes stable in a M dimensional space
(usually N>>M)
Different kinds of attractors: point, line, ring, plane, cyclic, chaotic …
y(t) activity of the network
e(t) external input
W,V weight matrices
Stable state for
Attractor networks
Stable state for
Trivial solution: I1=I2=0
Usually, attractor networks are symmetric, i.e.
With F(x) = [ x ]
+
we check whether there is a solution for I1>0 and I2>0
Stable solutions for v=1-w : I1=I2
v=0.2, w=0.8
v=0.1, w=0.8
Attractor networks
A specific application of firing rate models are continuous attractor
networks, which are used in the literature for various purposes.
Examples: short term memory, working memory, spatial navigation
in the head direction cell system and the entorhinal grid cell
system, coordinate transformation in the parietal cortex, gain
modulation in the pursuit system …
Continuous attractor networks are sometimes called neural fields.
The shape of the kernel wij determines the behaviour
of the network.
Common kernels are built of one Gaussian (blue) or
two Gaussians (green).
If the kernel becomes negative, inhibitory synapses
are required, and often modelled as two separate
kernels, one for inhibition and one for excitation.
Dale’s law: one neuron can make only excitatory or
only inhibitory synapses.
Attractor networks
excitatory units
inhibitory unit
One example is the proposed use of attractor
networks in short term memory for spatial
navigation, specifically in the head direction cell
system.
Sharp et al. 2001
Vi : net activation or average ‘membrane
potential’ of unit i
Ei : weighted input
Fi : firing rate
Si : synaptic drive
wij: synaptic weights
*i : position of unit i
Some properties of attractor networks
Continuous attractors can be supplied with
external input that is subsequently retained in
memory.
Bump shifts to the external
input and remains at that
position.
If the external input is
subsequently removed, the
population response of the
network retains its value.
Some properties of attractor networks
Continuous attractors are insensitive to noise.
Uniform input noise does not
change the shape of the bump.
Even if the external input is
almost hidden in noise, the
attractor network will still
detect it.
Some properties of attractor networks
Gain-modulated receptive field of a single unit in a network of 500 cells. The result (lines in A) is the
same as multiplying by fixed factors (squares in A). Input is shown in B. Thus, the relation between
input and bump height is multiplicative (Salinas & Abbott 1996). C and D show the outpus of two
types of feed-forward networks, which is not multiplicative.
Some properties of attractor networks
Continuous attractors can fuse external input in
various ways depending on the strength and
shape of the input.
Low input amplitude: The
bump shifts to the closer of
the two external inputs, even
though it is smaller.
For higher overall input
activity, the bump doesn’t
shift anymore, but switches
to the higher of the two
bumps.
The cognitive map
1978
Muller et al. 1987
O’Keefe & Dostrovsky 1971: Place cells in the rat hippocampus
Head direction cell network
Sharp et al. 2001
Head direction cells fire whenever the
animal’s head points to a specific spatial
direction. This works without vision, and is
dependent on an intact vestibular system.
Head direction pathway:
Tegmentum (DTN)
Mammillary bodies (LMN)
Thalamus (ADN)
Postsubiculum (Post)
Entorhinal cortex (EC)
The temporal lobe
The hippocampus and surrounding areas
Bird & Burgess 2008
The temporal lobe
The hippocampus and its connections
Bird & Burgess 2008
Head direction cell network
McNaughton et al. 2006
The simplest network consists of two recurrently connected
populations of neurons (rings), each of them receiving velocity
input (Xie et al. 2002). Due to the asymmetric connectivity between
rings, an initially stable “bump” of activity will move, if velocity input
is given to the rings.
Head direction cell network
Xie et al. 2002
A snapshot of the traveling bumps in two rings. The dashed lines
denote the synaptic variables and the solid lines represent the firing
rate.
Neural activity (f) and synaptic activation (s) profiles of two
rings in the stationary (a) and the moving state (b).
This model constitutes the minimal model achieving
both short-term memory and integration properties.
Head direction cell network
Goodridge & Touretzky 2000
The head direction cell system is supposed to perform
path integration, i.e., to convert self-velocity to selforientation.
It has been proposed that one or more regions of the HD
pathway contain continuous attractors.
Head direction cell network
ccw
cw
VN
LMN
AD
PoS
Goodridge & Touretzky 2000
The LMN layer receives input from angular velocity units.
The connections from LMN to AD are asymmetric, so that
during movement the two bumps from LMN result in an
asymmetric bump in AD, which shifts the bump in PoS.
LMN are PoS are modelled as continuous attractors.
Head direction cell network
Goodridge & Touretzky 2000
The LMN layer receives input from angular velocity units.
The connections from LMN to AD are asymmetric, so that
during movement the two bumps from LMN result in an
asymmetric bump in AD, which shifts the bump in PoS.
LMN are PoS are modelled as continuous attractors.
Head direction cell network
Boucheny et al. 2005
Since intrinsic recurrent connections in the head direction
cell network are questionable, Boucheny et al. proposed a
three-population network without recurrent connections.
What about distances and places?
Best et al. 2001
Leutgeb et al. 2005
In rats, so-called place cells have been found in the
hippocampus. These cells respond selectively when the rat
is at a certain position within the arena.
How to get a place cell “map”?
McNaughton et al. Nat Rev Neurosci 7, 2006
Leutgeb et al. Curr Opin Neurobiol 15, 2005
The group of Edvard Moser recently discovered the so-called grid
cells in the medial entorhinal cortex of the rat.
How to get place cells from grid cells
The group of Edvard Moser recently discovered the so-called grid
cells in the medial entorhinal cortex of the rat.
McNaughton et al. Nat Rev Neurosci 7, 2006
One dimensional simplification
simulated entorhinal
grid cells
simulated place cells
Each entorhinal grid cell layer can be conceived as oscillator driven
by self-motion. Place cells encode the relative phase of grid cell
populations.
Entorhinal grid cell and place cell simulation: each layer EN1-5 consists of a continuous attractor network of 300 rate neurons
(after Bucheny, Brunel, Arleo. J Comp Neurosci 18, 2005)
Place cell layer (n=100) receives a weighted linear combination of EN1-5
(see McNaughton et al. Nat Rev Neurosci 7, 2006)
What are place cells good for?
What are place cells good
for, if they only provide an
information about our
current location?
The hippocampus is the
crucial structure for
remembering the current
past (episodic memory).
Place cells might be the glue
between the current position
and other properties of this
position (e.g., an object,
which can be seen, or
another previous experience
at that place).
Bird & Burgess 2008
The standard place cell experiment
Revisiting the rat to learn for the human
Muller et al. 1987
Task: search for food pellets in the arena
Assumption: effective strategy requires remembering pellet locations
Rats perform a task possibly involving spatial memory
while place cells are recorded
Exploring with place cells
Can we use place cells for navigation and spatial exploration?
physical feedback
grid cell layer
current location
motor command
“place cell” layer
desired location
border cell layer
Basis for the model: a place-cell like attractor network in a closed sensorimotor loop.
Assumptions: Place cells indicate the desired rather than the current location.
Staying in one location increases the inhibitory synaptic weight, and will push the
neural activity away from that location.
Modelling explorative behaviour
Computational model of explorative behavior
Toroidal geometry as in McNaughton et al. 2006
Assumptions
‘Place cell’ activity indicates desired location rather than current location
Effective exploration leaves a memory trace of visited locations
Exploring with place cells
Can we use place cells for navigation and spatial exploration?
Simulated trajectory of an
agent during spatial
exploration of a closed
space.
The basis for the simulation
is place-cell like attractor
network, which is put in a
closed sensorimotor loop.
Modelling explorative behaviour
Model produces realistic exploratory behavior and simulates features of place cell
activity seen in rats
Visuo-spatial hemi-neglect
Kerkhoff 2001
After lesions of the right temporoparietal or ventral frontal cortex
Attention
Corbetta et al. 2008
The Dorsal and Ventral Attention Systems
‘Attention’ model
Network Outline
retina
MSO
MSO
V1
inhibitory
interneurons
superior
colliculus
Simple model for visuo-spatial neglect
recurrent
winner-takes-all
‘Attention’ model
left
two
right
target
retina
MSO
V1
SC
}
attention
left
}
no preference
}
right
Healthy system: distribution of attention to one hemisphere
‘Attention’ model
target
left
two
right
retina
MSO
V1
SC
}
neglect
}
healthy
Lesioned system: attention always in one hemisphere