Computational neuroscience and neuroinformatics
Mathematical models of neuron and synapse activity:
Neuron: 1011 neurons communicating via electric impulses via 10 15 synapses
Many neuron types bv pyramidal/principal neurons most dominant neurons
Neuron structure:
Dendrite: recieves input from other neurons
Soma: integrates input from neurons decides whether to generate an AP
Axon: propagation of AP to another neuron
AP/spikes: large deviation in electrical potential between the inside and outside of the neuron lasting very short
Rare events
Triggered at threshold
Subthreshold potential: little potential changes over time = resting potential
Interneurons: decrease / increase the electrical potential of a neuron
Model: does not represent the brain precisely but can be used to answer a specific research question
¿Neuron ≈ electric circuit ≈ math model
Simple neuron model:
Explain sub-threshold fluctuations
Passive membrane model:
Resistance capacitance circuit: simple representation of neuron
Capacitance (C ): charges deposited on the plates and since 2
plates are close together they get deposited on the second
C R plate with an opposite charge
Resitance ( R )
Battery (urest )
urest
Outside current I splits into I c and I R
UR u−urest
Ohm’s law: I R = → I R=
R R
Coulomb’s law: Q=C∗u = capacitor C stores charge Q depending on the voltage u
dQ C∗du
I C= =
dt dt
u−u rest C∗du
I =I R + I C = +
R dt
RC∗du
We’re interested in the potential change + simplify formula(¿ R ): =−( u−urest ) + RI
dt
dV du
V =u−urest = because urest is constant so d urest =0
dt dt
Time constant τ : determines how fast the potential will decay in the absence of a current τ =RC
, τ ↑: slow decay
τ ↓: fast decay (fast responding neuron)
dV
τ =−V + RI
dt
Constant current in passive membrane model:
= Step current of constant amplitude
du
Before t 0 : τ =−(u−u rest )
dt
du
Fixed point: when =0 so u=urest
dt
du
After t 0 : τ =−( u−urest ) + RI 0
dt
No potential change (because current is fixed) : u=urest + RI 0
τ : determines how fast urest will jump to urest + RI 0
Might not reach plateau depending on how slow τ is
If current is infinitely on you reach urest + RI 0
Function of slope: 1−e−(t−t 0 )/τ
Graph: u ( t )=urest + RI 0∗[ 1−e−(t−t )/ τ ] 0
Pulse current in passive membrane model:
Pulse current: I 0 applied for a very short fraction of time ∆
Until t 0+ ∆ : potential change is the same as in a step current
u ( t 0+ ∆ ) =urest + RI 0 [ 1−e ] ¿ urest + RI 0 [ 1−e−∆/ τ ]
−(t 0+ ∆−t 0 )/ τ
2 3
If ∆ ≪ τ : −x
(Taylor series: e
x x
=1−x + + + …)
2 ! 3!
, [ ( ( ) )]
2
−∆ ∆
u ( t 0+ ∆ ) =urest + RI 0 1− 1− + +…
τ τ
[
Ignore further terms because they are very small ( ∆ ≪ τ ) : u ( t 0+ ∆ ) =urest + RI 0 1−1+
∆
τ
+0 ∆ ²
]
RI 0 ∆
u ( t 0+ ∆ ) =urest +
τ
Charge:
Pulse current: very short current pulse doesn’t have the time to pass the resistors so gets reposited
on the capacitors
Charge deposited on capacitors: q=I 0∗∆ (pulse width halved but current doubled -> same charge)
q q
u=urest + the same for each pulse when the same current is applied
C C
Leaky integrate-and fire model:
Integrate and fire: passive membrane model + condition
Condition: if u = threshold θ a spike is fired and u gets reset to ur (= ureset )
Leaky: the potential doesn’t reset to ur instantly after reaching threshold
Step current in leaky integrate and fire model:
urest + RI 2
I2
I 1=I c urest + RI 1 θ
ur
Step current too low:
Potential will not reach the threshold and remain at urest + RI 0
Critical current I c reached:
Potential will reach threshold and go back to ur
Will increase again (because current is constant) and fire spikes repeatedly
Time to reach threshold: is a function of the current magnitude
f
, → As current increases the firing frequency increases
Ic I
Higher current:
Threshold will be reached faster
I =0 :
Slope: depends on τ
Intersection = urest = stable fixed point
Stable fixed point: if the potential is slightly off in either
direction it will aways go back to urest
Step current:
Constant value added so all values increase
slope doesn’t change
Intersection: would be after threshold θ
No fixed point because it would be after threshold
will never get reached because potential gets reset
ur to ur when reaching θ
Intersection just after θ :
du
As u approaches θ the rate of change becomes smaller and smaller
dt
Small firing frequency
= Type 1 neurons
Non-linear integrate and fire model:
= quadratic integrate and fire model
I =0 :
Fixed point 1 = stable fixed point: attracts values above and below
du
o Below fixed point 1: is positive so will increase
dt
until fixed point reached
du
o Above fixed point 1: is negative so will decrease
dt
towards the fixed point
Fixed point 2 = unstable fixed point:
o On fixed point: u will stay the same
du
o Below fixed point: is negative so will decrease
dt
towards stable fixed point
du
o Above fixed point: is positive so u increases until θ reached
dt
= effective threshold
Mathematical models of neuron and synapse activity:
Neuron: 1011 neurons communicating via electric impulses via 10 15 synapses
Many neuron types bv pyramidal/principal neurons most dominant neurons
Neuron structure:
Dendrite: recieves input from other neurons
Soma: integrates input from neurons decides whether to generate an AP
Axon: propagation of AP to another neuron
AP/spikes: large deviation in electrical potential between the inside and outside of the neuron lasting very short
Rare events
Triggered at threshold
Subthreshold potential: little potential changes over time = resting potential
Interneurons: decrease / increase the electrical potential of a neuron
Model: does not represent the brain precisely but can be used to answer a specific research question
¿Neuron ≈ electric circuit ≈ math model
Simple neuron model:
Explain sub-threshold fluctuations
Passive membrane model:
Resistance capacitance circuit: simple representation of neuron
Capacitance (C ): charges deposited on the plates and since 2
plates are close together they get deposited on the second
C R plate with an opposite charge
Resitance ( R )
Battery (urest )
urest
Outside current I splits into I c and I R
UR u−urest
Ohm’s law: I R = → I R=
R R
Coulomb’s law: Q=C∗u = capacitor C stores charge Q depending on the voltage u
dQ C∗du
I C= =
dt dt
u−u rest C∗du
I =I R + I C = +
R dt
RC∗du
We’re interested in the potential change + simplify formula(¿ R ): =−( u−urest ) + RI
dt
dV du
V =u−urest = because urest is constant so d urest =0
dt dt
Time constant τ : determines how fast the potential will decay in the absence of a current τ =RC
, τ ↑: slow decay
τ ↓: fast decay (fast responding neuron)
dV
τ =−V + RI
dt
Constant current in passive membrane model:
= Step current of constant amplitude
du
Before t 0 : τ =−(u−u rest )
dt
du
Fixed point: when =0 so u=urest
dt
du
After t 0 : τ =−( u−urest ) + RI 0
dt
No potential change (because current is fixed) : u=urest + RI 0
τ : determines how fast urest will jump to urest + RI 0
Might not reach plateau depending on how slow τ is
If current is infinitely on you reach urest + RI 0
Function of slope: 1−e−(t−t 0 )/τ
Graph: u ( t )=urest + RI 0∗[ 1−e−(t−t )/ τ ] 0
Pulse current in passive membrane model:
Pulse current: I 0 applied for a very short fraction of time ∆
Until t 0+ ∆ : potential change is the same as in a step current
u ( t 0+ ∆ ) =urest + RI 0 [ 1−e ] ¿ urest + RI 0 [ 1−e−∆/ τ ]
−(t 0+ ∆−t 0 )/ τ
2 3
If ∆ ≪ τ : −x
(Taylor series: e
x x
=1−x + + + …)
2 ! 3!
, [ ( ( ) )]
2
−∆ ∆
u ( t 0+ ∆ ) =urest + RI 0 1− 1− + +…
τ τ
[
Ignore further terms because they are very small ( ∆ ≪ τ ) : u ( t 0+ ∆ ) =urest + RI 0 1−1+
∆
τ
+0 ∆ ²
]
RI 0 ∆
u ( t 0+ ∆ ) =urest +
τ
Charge:
Pulse current: very short current pulse doesn’t have the time to pass the resistors so gets reposited
on the capacitors
Charge deposited on capacitors: q=I 0∗∆ (pulse width halved but current doubled -> same charge)
q q
u=urest + the same for each pulse when the same current is applied
C C
Leaky integrate-and fire model:
Integrate and fire: passive membrane model + condition
Condition: if u = threshold θ a spike is fired and u gets reset to ur (= ureset )
Leaky: the potential doesn’t reset to ur instantly after reaching threshold
Step current in leaky integrate and fire model:
urest + RI 2
I2
I 1=I c urest + RI 1 θ
ur
Step current too low:
Potential will not reach the threshold and remain at urest + RI 0
Critical current I c reached:
Potential will reach threshold and go back to ur
Will increase again (because current is constant) and fire spikes repeatedly
Time to reach threshold: is a function of the current magnitude
f
, → As current increases the firing frequency increases
Ic I
Higher current:
Threshold will be reached faster
I =0 :
Slope: depends on τ
Intersection = urest = stable fixed point
Stable fixed point: if the potential is slightly off in either
direction it will aways go back to urest
Step current:
Constant value added so all values increase
slope doesn’t change
Intersection: would be after threshold θ
No fixed point because it would be after threshold
will never get reached because potential gets reset
ur to ur when reaching θ
Intersection just after θ :
du
As u approaches θ the rate of change becomes smaller and smaller
dt
Small firing frequency
= Type 1 neurons
Non-linear integrate and fire model:
= quadratic integrate and fire model
I =0 :
Fixed point 1 = stable fixed point: attracts values above and below
du
o Below fixed point 1: is positive so will increase
dt
until fixed point reached
du
o Above fixed point 1: is negative so will decrease
dt
towards the fixed point
Fixed point 2 = unstable fixed point:
o On fixed point: u will stay the same
du
o Below fixed point: is negative so will decrease
dt
towards stable fixed point
du
o Above fixed point: is positive so u increases until θ reached
dt
= effective threshold
