Summary Computational
Neuroscience
Week 1: what we need and don’t need to know:
Biophysics and mathematics need to know:
- What a differential equation is.
- How to calculate the steady state values for a simple differential equation.
- How to test if the given solution of a differential equation is in line with the calculated steady state
values.
- The two components of a neuronal membrane that together produce the electrical behavior of a
passive membrane
- Positive current injection leads to membrane depolarization
- Negative current injection leads to membrane hyperpolarization
Don’t need to know:
- How to analytically solve a differential equation
- To create a differential equation for a given system
- Remember the formula’s for capacitive and the current through a ion channel
- The exact formulation of the Hodgkin-Huxley equations
Live neural network simulation need to know:
- All concepts that are explained in the PPT
- Know how to calculate whether a neuron will be active given the neuronal input, synaptic weights and
threshold
Neuron practical
- understand the general concepts and principles of excitability.
- can predict for very obvious manipulations what happens if a certain stimulus is given or a model
parameter is changed
- Network tutorial 2 is NOT part of the exam
Lecture: introduction into computational neuroscience
Computational neuroscience is a branch of neuroscience which employs mathematical models, computer
simulations, theoretical analysis and abstractions of the brain the understand the principles that govern the
development, structure, physiology and cognitive abilities of the nervous system. It’s also know as theoretical
neuroscience or mathematical neuroscience. Equations force a model to be precise, complete and self-
consistent. It’s a fast way to generate and vet ideas prior to full experimental testing. The key test of the value
of a theory is not necessarily whether it predicts something new, but whether it produces concepts that
generalize to other systems and provide valuable new ways of thinking. A good model has just enough
complexity to understand the phenomena you want to describe. A good model is if it produces emerging
properties, that are not explicitly modeled in the individual components but arise from interaction of the
different elements. You should chose the right level of detail and complexity based on what you want to know
from the model. Brian models could be about excitability, AP propagation, connectivity, memory and brain
disease.
Properties and interactions of components at a lower level determine the behaviour (emergent properties) of
the system one level up. Example: electrical properties of individual neurons and their synaptic connections
determine the frequency of oscillations (emergent property) observed in neuronal networks.
With this approach we can try to understand how patient mutations lead to brain disease. Leaky-Integrate-and-
Fire models are efficient neuron models to simulate activity in large networks.
Lecture: live neuron simulation
In the live simulation, we worked as a Mc Cullock-Pitts neuron model. The input can be 0 or
1, active or active. The weights are the synaptic strength of the neurons. The synapse is
,strong when there are more receptors active post-synaptically. The threshold is the firing threshold for the
action potential.
To see whether the neuron is active, the model has two steps:
1.
2.
Each neuron has a unique set of synaptic weights with which it’s connected to the neurons in the
next layer. The synaptic weights determine the computational task that is performed by the
network.
The take home message: neurons have special properties which equips them for information processing in a
network configuration. A neural network is able to make more complex computations than on the single
neuron level. Synaptic weights determine the functionality of the network. By adapting synaptic weights, the
brain is able to learn new things! Wow!
Lecture: differential equations
- Many different biological processes can be modelled with differential equations.
- Analytical solutions only exist for a limited set of models
- Numerical methods can be used to solve models that do not have an analytical solution
- Steady state solutions can always be calculated if they exist
- The membrane capacitance and leak-channels together determine the passive electrical properties of
a cell.
- Injecting a current depolarizes the cell.
- The steady state voltage in determined by the membrane resistance and the amplitude of the injected
current.
- Leaky-Integrate-and-Fire models are efficient neuron models to simulate activity in large networks.
Neuron models: how do we model the brain?
1. Decide what aspects of the brain you want to simulate
Electrical signaling
i. Injecting negative current: hyperpolarization of membrane
ii. Injecting positive current: depolarization of membrane, when threshold exceeded,
action potential will be fired.
2. Choose the right level of complexity
Use of the point neuron, simulated by an electrode. Less parameters!!
3. Define a set of rules that might explain your observations
Lipid bilayer is impermeable for ions. The amount of potential difference between the
outside and inside is given by the amount of charge. Ion channels are embedded in the
membrane and act as a gate or channel.
Electrical current in movement of electrical charge
Ionic movement is the electrical current due to movement of ions.
The more open ion channels: the lower the membrane resistance
4. Formulate these rules in mathematical formulas
V= membrane potential
I= ionic current
Q= charge
G= conductance
C= capacitance
dV
: change of the membrane potential over time
dT
, dV
=0 Steady state is reached, when dV/dT is zero, thus the slope of the graph must be 0.
dT
Integration ∫ is the opposite of the derivative. When derivative is taken, integration explains how the initial
function changes over time.
Synaptic signaling: injecting current in the postsynaptic membrane. Receptors are ion channels activated
through the binding of neurotransmitter. Signalling occurs due to a summation of synaptic potentials.
5. Implement these in software and hardware
6. Run simulations to compare with real data
7. Make testable predictions with the model
8. Test model predictions with experiments
9. Repeat
Lecture: modelling ion channels
Electrical current is the movement of electrical charges. Ionic current is the electric current due to the
movement of ions.
The lipid bilayer acts as capacitor. Charge can only goa cross the membrane via ion channels. These ion
channels are conductance and follow Ohms law.
This differential equation on the right describes how the membrane potential changes.
Possible exam question: differential equation is given and steady state must be solved.
Steady state is dV/dT=0, so the expression of the steady state value is V ss=EL+RmIe
The differential equation doesn’t need to be solved analytically. However:
When you’re given V= (V0-Vss)e-kt + Vss, test whether this equation is in line with the value
that is calculated for the steady state. How?
- In E-kt, the t becomes infinite, e becomes so small, it can be cancelled out. You’re left with (V 0-Vss) + Vss.
o E0=1
This is a AMPA receptor that binds glutamate. When Na + enters the cell, there is EPSC: excitatory
postsynaptic current. This leads to depolarization of the membrane and an EPSC.
This is a GABA receptor that binds GABA. When Cl- enters the cell, there is IPSC: inhibitory
postsynaptic current. This leads to hyperpolarization of the membrane and IPSC.
Most of the times one active synapse cannot stimulate enough for a threshold to fire. This
usually takes more synapses. If only the inhibitory synapse is active, hyperpolarization
occurs. If all three neurons are active, the inhibitory synapse can be strong enough to
prevent firing of AP. In the figure, synapse E1 and E2 are excitatory and synapse I is
inhibitory.
Leaky-Integrate-and-Fire model (Louis Lapique 1907), describes when neurons are active. Depending on the
synaptic inputs that arrive at the cell, the threshold will be excited and the neuron will fire AP. The LIF models
defines APs as reached or not reached. This makes models about the brains much easier to simulate, an
advantage of the LIF models. This model thus, doesn’t say anything about how much the threshold is or isn’t
exceeded. At the time of making this model, they knew that there were spiked and threshold, however not
how and why they were.
LIF model computer practical:
Neuroscience
Week 1: what we need and don’t need to know:
Biophysics and mathematics need to know:
- What a differential equation is.
- How to calculate the steady state values for a simple differential equation.
- How to test if the given solution of a differential equation is in line with the calculated steady state
values.
- The two components of a neuronal membrane that together produce the electrical behavior of a
passive membrane
- Positive current injection leads to membrane depolarization
- Negative current injection leads to membrane hyperpolarization
Don’t need to know:
- How to analytically solve a differential equation
- To create a differential equation for a given system
- Remember the formula’s for capacitive and the current through a ion channel
- The exact formulation of the Hodgkin-Huxley equations
Live neural network simulation need to know:
- All concepts that are explained in the PPT
- Know how to calculate whether a neuron will be active given the neuronal input, synaptic weights and
threshold
Neuron practical
- understand the general concepts and principles of excitability.
- can predict for very obvious manipulations what happens if a certain stimulus is given or a model
parameter is changed
- Network tutorial 2 is NOT part of the exam
Lecture: introduction into computational neuroscience
Computational neuroscience is a branch of neuroscience which employs mathematical models, computer
simulations, theoretical analysis and abstractions of the brain the understand the principles that govern the
development, structure, physiology and cognitive abilities of the nervous system. It’s also know as theoretical
neuroscience or mathematical neuroscience. Equations force a model to be precise, complete and self-
consistent. It’s a fast way to generate and vet ideas prior to full experimental testing. The key test of the value
of a theory is not necessarily whether it predicts something new, but whether it produces concepts that
generalize to other systems and provide valuable new ways of thinking. A good model has just enough
complexity to understand the phenomena you want to describe. A good model is if it produces emerging
properties, that are not explicitly modeled in the individual components but arise from interaction of the
different elements. You should chose the right level of detail and complexity based on what you want to know
from the model. Brian models could be about excitability, AP propagation, connectivity, memory and brain
disease.
Properties and interactions of components at a lower level determine the behaviour (emergent properties) of
the system one level up. Example: electrical properties of individual neurons and their synaptic connections
determine the frequency of oscillations (emergent property) observed in neuronal networks.
With this approach we can try to understand how patient mutations lead to brain disease. Leaky-Integrate-and-
Fire models are efficient neuron models to simulate activity in large networks.
Lecture: live neuron simulation
In the live simulation, we worked as a Mc Cullock-Pitts neuron model. The input can be 0 or
1, active or active. The weights are the synaptic strength of the neurons. The synapse is
,strong when there are more receptors active post-synaptically. The threshold is the firing threshold for the
action potential.
To see whether the neuron is active, the model has two steps:
1.
2.
Each neuron has a unique set of synaptic weights with which it’s connected to the neurons in the
next layer. The synaptic weights determine the computational task that is performed by the
network.
The take home message: neurons have special properties which equips them for information processing in a
network configuration. A neural network is able to make more complex computations than on the single
neuron level. Synaptic weights determine the functionality of the network. By adapting synaptic weights, the
brain is able to learn new things! Wow!
Lecture: differential equations
- Many different biological processes can be modelled with differential equations.
- Analytical solutions only exist for a limited set of models
- Numerical methods can be used to solve models that do not have an analytical solution
- Steady state solutions can always be calculated if they exist
- The membrane capacitance and leak-channels together determine the passive electrical properties of
a cell.
- Injecting a current depolarizes the cell.
- The steady state voltage in determined by the membrane resistance and the amplitude of the injected
current.
- Leaky-Integrate-and-Fire models are efficient neuron models to simulate activity in large networks.
Neuron models: how do we model the brain?
1. Decide what aspects of the brain you want to simulate
Electrical signaling
i. Injecting negative current: hyperpolarization of membrane
ii. Injecting positive current: depolarization of membrane, when threshold exceeded,
action potential will be fired.
2. Choose the right level of complexity
Use of the point neuron, simulated by an electrode. Less parameters!!
3. Define a set of rules that might explain your observations
Lipid bilayer is impermeable for ions. The amount of potential difference between the
outside and inside is given by the amount of charge. Ion channels are embedded in the
membrane and act as a gate or channel.
Electrical current in movement of electrical charge
Ionic movement is the electrical current due to movement of ions.
The more open ion channels: the lower the membrane resistance
4. Formulate these rules in mathematical formulas
V= membrane potential
I= ionic current
Q= charge
G= conductance
C= capacitance
dV
: change of the membrane potential over time
dT
, dV
=0 Steady state is reached, when dV/dT is zero, thus the slope of the graph must be 0.
dT
Integration ∫ is the opposite of the derivative. When derivative is taken, integration explains how the initial
function changes over time.
Synaptic signaling: injecting current in the postsynaptic membrane. Receptors are ion channels activated
through the binding of neurotransmitter. Signalling occurs due to a summation of synaptic potentials.
5. Implement these in software and hardware
6. Run simulations to compare with real data
7. Make testable predictions with the model
8. Test model predictions with experiments
9. Repeat
Lecture: modelling ion channels
Electrical current is the movement of electrical charges. Ionic current is the electric current due to the
movement of ions.
The lipid bilayer acts as capacitor. Charge can only goa cross the membrane via ion channels. These ion
channels are conductance and follow Ohms law.
This differential equation on the right describes how the membrane potential changes.
Possible exam question: differential equation is given and steady state must be solved.
Steady state is dV/dT=0, so the expression of the steady state value is V ss=EL+RmIe
The differential equation doesn’t need to be solved analytically. However:
When you’re given V= (V0-Vss)e-kt + Vss, test whether this equation is in line with the value
that is calculated for the steady state. How?
- In E-kt, the t becomes infinite, e becomes so small, it can be cancelled out. You’re left with (V 0-Vss) + Vss.
o E0=1
This is a AMPA receptor that binds glutamate. When Na + enters the cell, there is EPSC: excitatory
postsynaptic current. This leads to depolarization of the membrane and an EPSC.
This is a GABA receptor that binds GABA. When Cl- enters the cell, there is IPSC: inhibitory
postsynaptic current. This leads to hyperpolarization of the membrane and IPSC.
Most of the times one active synapse cannot stimulate enough for a threshold to fire. This
usually takes more synapses. If only the inhibitory synapse is active, hyperpolarization
occurs. If all three neurons are active, the inhibitory synapse can be strong enough to
prevent firing of AP. In the figure, synapse E1 and E2 are excitatory and synapse I is
inhibitory.
Leaky-Integrate-and-Fire model (Louis Lapique 1907), describes when neurons are active. Depending on the
synaptic inputs that arrive at the cell, the threshold will be excited and the neuron will fire AP. The LIF models
defines APs as reached or not reached. This makes models about the brains much easier to simulate, an
advantage of the LIF models. This model thus, doesn’t say anything about how much the threshold is or isn’t
exceeded. At the time of making this model, they knew that there were spiked and threshold, however not
how and why they were.
LIF model computer practical:
