COMPUTATIONAL NEUROSCIENCES
1. Introduction, Overview & Foundations of Neurodynamics
Introduction
Why study brain ?
• Brain is probably the most complex and yet interesting organ of the body.
• Fundamental understanding of structure, interaction between different parts and the function
of the brain.
• Brain is a high-dimensional complex network.
o Focal stroke, not just the output of the directly affected neurons will be disturbed
o Not: 1 brain region - 1 function
• Several brain disorders have now been identified
o Epilepsy, stroke, neurodegeneration, Schizophrenia, Autism that need to be better
understood to identify treatment strategies.
• Brain can be studied at several scales: molecular, cellular, microcircuit, population, and
system level to behaviour.
• Opens up possibilities to develop novel methods of probing the brain at different scales as well
as analysing complex datasets. (e.g. optogenetics)
Complexity of the brain
• Brain is made up of predominantly neurons, but also microglia
o Microglia: energy supply & structural stabilization of brain tissue
• 1011 neurons; 1015 synapses/connections; each neuron receives ~10000 synapses from other
neuron.
• Many different types of neurons exist in terms of size, shape and molecular properties.
• Neurons communicate via electrical impulses, called action potentials.
o Frequency and rate will depend on type of neuron and input it receives
Overview
Overview: Neurons
- Dendrites: ‘input device’ receives input from other neurons, transmit them to the soma
- Soma; ‘central processing unit’: integrates info by nonlinear processing step, if total input
arriving at soma exceeds the threshold, then an output signal is generated
- Axon: ‘output device’: ~wire, carries electrical signal to other neurons
1
,Overview generation of an AP
- Different concentrations on both sides of the cell → ° potential difference (=membrane potential)
- When potential differences decreases to certain level → neuron fires
o Depolarization will depend on the input the neuron receives from other neurons
- Neural signal of a single neuron consists of short electrical pulses (spike train)
o Each pulse = AP/spike
o AP amplitude = 100mV, duration = 1-2ms
o Form of the AP does not change
Overview: simple Neuron Model
Overview: Spikes and Subtreshold regime
• Output
o Spikes= AP are rare events
▪ Exc. Bursting neurons fire more spikes at a time (=interneurons?) but afterwards
they will be silent, and the potential will be subthreshold again
o Are triggered at tresholds
• Below threshold = subthreshold regime
o The membrane potential fluctuates, if it reaches a threshold it fires an AP
2
, - Subthreshold fluctuations before AP
Foundations of Neurodynamics
1.1.1. A simple Neuron Model
The passive membrane
• The passive membrane doesn’t generate spikes
• Focus on subthreshold regime, Everything is linear
• The simplest model of a passive membrane = RC circuit
o R = resting membrane resistance + intracellular axial resistance along axons & dendrites
o C= membrane capacitance (in parallel with membrane resistance)e
➔ 3 passive electrical properties of neurons
• Active membrane responses = responses that occur whenever ion channels are gated by
channels r chemicals
Fig 1: the EPSP caused by the arrival of a spike from
neuron jj at an excitatory synapse of neuron ii.
The cell membrane acts like a
capacitor in parallel with a resistor
which is in line with a battery of
potential Urest (zoomed inset).
If the driving force vanishes, the
voltage across the capacitor is
given by the battery voltage urest
See movies Neuronal Dynamics
3
, -------------------------------------------------------------------------
Post-synapticpotential
❖ The timecourse of ui (t) of the membrane potential of neuron i
• With electrode we can measure the potential difference u(t) between in & out = membrane
potential
o Without input → neuron is at rest → constant membrane potential urest
• Before the input ui(t)=urest .
• At t=0 the presynaptic neuron j fires its spike. For t>0, we see at the electrode a response of
neuron i arrives
𝑢𝑖 (𝑡) − 𝑢𝑟𝑒𝑠𝑡 =: ∈𝑖𝑗 (𝑡)
o The right part of the equation defines the postsynaptic potential (PSP
• If the voltage difference 𝑢𝑖 (𝑡) − 𝑢𝑟𝑒𝑠𝑡 is positive/ negative we have an excitatory/inhibitory
postsynaptic potential, EPSP/ IPSP
See figure 1
--------------------------------------------------------------------------
Can we describe u(t) in response to/ in function of an input current I(t)?
❖ U(t) for an input I(t)
• The input current I(t) (coms from another neuron) gets divided over the capacitor & the resistor :
𝑰 = 𝑰𝑪 + 𝑰𝑹
o 𝑰𝑪 ?
𝑄
▪ 𝐶=
𝑢
Capacitor = constant,
𝑄
▪ 𝑈= Q = charge over the capacitance, will change as the current comes in
𝐶
𝑑𝑢 𝐼 𝑑𝑄
▪ = dq/dt = I
𝑑𝑡 𝐶 𝑑𝑡
𝑑𝑢 𝐼𝑐
▪ =
𝑑𝑡 𝐶
𝑑𝑢
▪ 𝐼𝑐 = 𝐶 ∗
𝑑𝑡
o 𝑰𝑹 ?
(𝑢−𝑢𝑟𝑒𝑠𝑡 )
▪ 𝐼𝑅 = Ohm’s Law: V=IR
𝑅
o 𝑰 = 𝑰𝑪 + 𝑰𝑹
𝑑𝑢 (𝑢−𝑢𝑟𝑒𝑠𝑡 )
▪ 𝐼= 𝐶∗ +
𝑑𝑡 𝑅
𝐶𝑑𝑢 −(𝑢−𝑢𝑟𝑒𝑠𝑡 )
= +𝐼
𝑑𝑡 𝑅
𝑑𝑢
𝑅𝐶 = −(𝑢 − 𝑢𝑟𝑒𝑠𝑡 ) + 𝑅𝐼
𝑑𝑡
= Equation of a passive membrane
= Linear Ordinary Differential equation
= RC equation to membrane potential changes as a function of the input current
4
1. Introduction, Overview & Foundations of Neurodynamics
Introduction
Why study brain ?
• Brain is probably the most complex and yet interesting organ of the body.
• Fundamental understanding of structure, interaction between different parts and the function
of the brain.
• Brain is a high-dimensional complex network.
o Focal stroke, not just the output of the directly affected neurons will be disturbed
o Not: 1 brain region - 1 function
• Several brain disorders have now been identified
o Epilepsy, stroke, neurodegeneration, Schizophrenia, Autism that need to be better
understood to identify treatment strategies.
• Brain can be studied at several scales: molecular, cellular, microcircuit, population, and
system level to behaviour.
• Opens up possibilities to develop novel methods of probing the brain at different scales as well
as analysing complex datasets. (e.g. optogenetics)
Complexity of the brain
• Brain is made up of predominantly neurons, but also microglia
o Microglia: energy supply & structural stabilization of brain tissue
• 1011 neurons; 1015 synapses/connections; each neuron receives ~10000 synapses from other
neuron.
• Many different types of neurons exist in terms of size, shape and molecular properties.
• Neurons communicate via electrical impulses, called action potentials.
o Frequency and rate will depend on type of neuron and input it receives
Overview
Overview: Neurons
- Dendrites: ‘input device’ receives input from other neurons, transmit them to the soma
- Soma; ‘central processing unit’: integrates info by nonlinear processing step, if total input
arriving at soma exceeds the threshold, then an output signal is generated
- Axon: ‘output device’: ~wire, carries electrical signal to other neurons
1
,Overview generation of an AP
- Different concentrations on both sides of the cell → ° potential difference (=membrane potential)
- When potential differences decreases to certain level → neuron fires
o Depolarization will depend on the input the neuron receives from other neurons
- Neural signal of a single neuron consists of short electrical pulses (spike train)
o Each pulse = AP/spike
o AP amplitude = 100mV, duration = 1-2ms
o Form of the AP does not change
Overview: simple Neuron Model
Overview: Spikes and Subtreshold regime
• Output
o Spikes= AP are rare events
▪ Exc. Bursting neurons fire more spikes at a time (=interneurons?) but afterwards
they will be silent, and the potential will be subthreshold again
o Are triggered at tresholds
• Below threshold = subthreshold regime
o The membrane potential fluctuates, if it reaches a threshold it fires an AP
2
, - Subthreshold fluctuations before AP
Foundations of Neurodynamics
1.1.1. A simple Neuron Model
The passive membrane
• The passive membrane doesn’t generate spikes
• Focus on subthreshold regime, Everything is linear
• The simplest model of a passive membrane = RC circuit
o R = resting membrane resistance + intracellular axial resistance along axons & dendrites
o C= membrane capacitance (in parallel with membrane resistance)e
➔ 3 passive electrical properties of neurons
• Active membrane responses = responses that occur whenever ion channels are gated by
channels r chemicals
Fig 1: the EPSP caused by the arrival of a spike from
neuron jj at an excitatory synapse of neuron ii.
The cell membrane acts like a
capacitor in parallel with a resistor
which is in line with a battery of
potential Urest (zoomed inset).
If the driving force vanishes, the
voltage across the capacitor is
given by the battery voltage urest
See movies Neuronal Dynamics
3
, -------------------------------------------------------------------------
Post-synapticpotential
❖ The timecourse of ui (t) of the membrane potential of neuron i
• With electrode we can measure the potential difference u(t) between in & out = membrane
potential
o Without input → neuron is at rest → constant membrane potential urest
• Before the input ui(t)=urest .
• At t=0 the presynaptic neuron j fires its spike. For t>0, we see at the electrode a response of
neuron i arrives
𝑢𝑖 (𝑡) − 𝑢𝑟𝑒𝑠𝑡 =: ∈𝑖𝑗 (𝑡)
o The right part of the equation defines the postsynaptic potential (PSP
• If the voltage difference 𝑢𝑖 (𝑡) − 𝑢𝑟𝑒𝑠𝑡 is positive/ negative we have an excitatory/inhibitory
postsynaptic potential, EPSP/ IPSP
See figure 1
--------------------------------------------------------------------------
Can we describe u(t) in response to/ in function of an input current I(t)?
❖ U(t) for an input I(t)
• The input current I(t) (coms from another neuron) gets divided over the capacitor & the resistor :
𝑰 = 𝑰𝑪 + 𝑰𝑹
o 𝑰𝑪 ?
𝑄
▪ 𝐶=
𝑢
Capacitor = constant,
𝑄
▪ 𝑈= Q = charge over the capacitance, will change as the current comes in
𝐶
𝑑𝑢 𝐼 𝑑𝑄
▪ = dq/dt = I
𝑑𝑡 𝐶 𝑑𝑡
𝑑𝑢 𝐼𝑐
▪ =
𝑑𝑡 𝐶
𝑑𝑢
▪ 𝐼𝑐 = 𝐶 ∗
𝑑𝑡
o 𝑰𝑹 ?
(𝑢−𝑢𝑟𝑒𝑠𝑡 )
▪ 𝐼𝑅 = Ohm’s Law: V=IR
𝑅
o 𝑰 = 𝑰𝑪 + 𝑰𝑹
𝑑𝑢 (𝑢−𝑢𝑟𝑒𝑠𝑡 )
▪ 𝐼= 𝐶∗ +
𝑑𝑡 𝑅
𝐶𝑑𝑢 −(𝑢−𝑢𝑟𝑒𝑠𝑡 )
= +𝐼
𝑑𝑡 𝑅
𝑑𝑢
𝑅𝐶 = −(𝑢 − 𝑢𝑟𝑒𝑠𝑡 ) + 𝑅𝐼
𝑑𝑡
= Equation of a passive membrane
= Linear Ordinary Differential equation
= RC equation to membrane potential changes as a function of the input current
4
