week 8, neuronal dynamics Flashcards
(25 cards)
what is a differential equation in terms of the solution
Differential equations are equations where the solution is not a single number, but instead the solution is a function of time f(t).
So it would be a bunch of numbers about how something changes in time
How to solve equations
- Analytical using a pen and paper
- Numerical using a computer. Thats what we do.
Usually when things are complex you only have the numerical, approximate solution. Comp neuro is almost always finding approximate solutions
what is the leaky integrate and fire model
you model the membrane potential and the summation of firing
Modelling membrane potential
Membrane potential:
Potential V is due to a difference in concentration of charged ions inside and outside the cell.
It can be increased by input currents from other neurons
Ohms law:
Voltage = Current (I) x Resistance (R)
Neuronal firing as summation:
Postsynaptic neuron receiving two presynaptic inputs (j)
Neuron firing is a summation
What is the integrate and fire model?
Two ingredients:
1. A differential equation describing evolution of the membrane potential
2. A mechanism to generate spikesW
what is added to the leaky integrate and fire model
- Charge slowly leaks through the membrane. This is known as leak resistance R.
how to compare neural models to empirical data
You could compare to firing rate
Or you could compare to inter-spike interval (1/firing-rate)
You can consider different spiking patterns as vectors in high dimensional space and compute the notion of distance between them. There are papers describing distance specifically between different spiking trains.
What are the two core components of the Integrate-and-Fire neuron model?
- An equation that describes how the membrane potential of the neuron evolves over time (usually a differential equation). 2. A spike-generation rule: when the membrane potential reaches a defined threshold, a spike is emitted, and the potential is reset to a resting value.
What does the Leaky Integrate-and-Fire (LIF) model add to the basic IF model?
The LIF model introduces a ‘leak’ term to simulate the passive decay of the membrane potential. This makes the neuron model more biologically realistic by accounting for the fact that the membrane potential tends to return to a resting level over time in the absence of input.
What happens when the membrane potential in the IF model reaches threshold?
When the membrane potential reaches the threshold, the model neuron emits a spike. Immediately after, the membrane potential is reset to a lower value, typically the resting potential. This mimics the firing and recovery behaviour of biological neurons.
Why are differential equations used in neuronal modelling?
Differential equations are used because they describe how variables (like membrane potential) change over time. This is essential in neuroscience where dynamic processes—such as the integration of inputs and generation of action potentials—are fundamental.
What is the role of a capacitor in the IF model?
The capacitor represents the neuron’s membrane, which stores electrical charge. It determines how quickly the membrane potential changes in response to input currents, influencing the temporal integration of synaptic inputs.
How does increasing input current affect the firing rate in LIF neurons?
Increasing the input current increases the rate at which the membrane potential approaches the threshold. As a result, the neuron fires more frequently, thus increasing its firing rate.
What is the Exponential Leaky Integrate-and-Fire (ELIF) model?
The ELIF model is an extension of the LIF model that includes an exponential term in the membrane potential equation. This term models the rapid upstroke of the action potential more realistically, capturing the steep rise in voltage during spike initiation.
What is the Adaptive Exponential Integrate-and-Fire (AdEx) model?
The AdEx model adds an adaptation current ‘w’ to the ELIF model. This current increases after each spike, making the neuron less likely to spike again immediately. It allows the model to reproduce features like spike-frequency adaptation and bursting observed in real neurons.
What do AdEx models capture that LIF and ELIF do not?
AdEx models can reproduce a wider variety of neuronal firing patterns, including adapting spike trains, bursting, and irregular spiking. This makes them more versatile for simulating realistic neuronal behaviour.
Name three key summary statistics used to evaluate neuronal models.
- Firing rate: how often a neuron fires over time. 2. Inter-spike interval: the time between consecutive spikes. 3. Spike timing: the exact timing of spikes, which is critical for information encoding.
What software package is recommended for simulating spiking neurons?
Brian 2 is recommended for simulating spiking neuron models. It is a flexible and user-friendly simulator built for Python, ideal for small- to medium-scale simulations. It can be installed using the command: pip install brian2
What is the role of neuronal models in neural engineering?
Neuronal models help interpret or generate neural signals in neuroprosthetic systems. For example, they can translate pressure from a robotic limb into spike trains that stimulate sensory nerves, providing touch feedback to amputees.
Leaky integrate and fire terms
Term Meaning Biological Interpretation
𝜏
𝑚
⋅
𝑑
𝑢
𝑑
𝑡
τ
m
⋅
dt
du
Rate of change of the membrane potential How quickly the neuron’s voltage is changing. This is the main thing we’re solving for.
𝜏
𝑚
τ
m
Membrane time constant:
𝑅
⋅
𝐶
R⋅C Determines how fast the membrane responds to inputs (i.e., how fast it leaks back to rest). Larger
𝜏
𝑚
τ
m
= slower change.
𝑑
𝑢
𝑑
𝑡
dt
du
Time derivative of membrane potential Tells how fast voltage is increasing or decreasing at any moment.
−
[
𝑢
(
𝑡
)
−
𝑢
rest
]
−[u(t)−u
rest
] Leak term The passive decay of voltage toward the resting potential. If
𝑢
(
𝑡
)
>
𝑢
rest
u(t)>u
rest
, this term is negative (voltage leaks down).
𝑅
⋅
𝐼
(
𝑡
)
R⋅I(t) Input term The effect of incoming synaptic or injected current. Larger input current pushes
𝑢
(
𝑡
)
u(t) upward.w
what does the exponential term do in the exponential fire equation
Δ T
: controls the sharpness of spike initiation (voltage steepness).Small Δ𝑇Δ T
: spike initiates suddenlyLarger Δ𝑇Δ T
: smoother ramp-up𝜗𝑟ℎϑ rh
: the rheobase threshold — the minimum voltage needed to initiate spiking with constant input
This term creates a gradual inflection in the voltage trace — just like in real neurons, where spikes aren’t infinitely sharp.
Describe adaptive leaky integrate and fire
The AdEx model builds on the ELIF model by introducing an adaptation current that changes over time. This adaptation variable increases each time the neuron fires and then slowly decays, creating a form of spike-triggered fatigue. As a result, AdEx can simulate more realistic neuronal behaviours like:
Spike-frequency adaptation (slower firing over time),
Bursting (clusters of rapid spikes), and
Irregular spiking patterns.
This model better captures how real neurons respond to sustained input, where firing rates are not constant. The adaptation depends on properties of ion channels and neuronal anatomy and is modelled using a second differential equation coupled to membrane voltage.
What happens when the adaption current W increases after a neuron spike
When the adaptation current increases after a spike, it causes a hyperpolarising effect on the membrane potential — in simple terms, it pulls the voltage down, making it harder for the neuron to spike again immediately.
