Chapter 3
Dynamics
3.1 Introduction to Dynamical Systems
Dynamical systems theory provides a powerful tool for analyzing nonlinear systems
of differential equations, including those that arise in neuroscience. This theory allows us to interpret solutions
3
Neural Decoding
3.1
Encoding and Decoding
In chapters 1 and 2, we considered the problem of predicting neural responses to known stimuli. The nervous system faces the reverse problem,
determining what is going on in the real world from neuronal spiking
Math 3375
MATH 1370 - Assignment 3
1. Synaptic balance and uctuation driven activity
In this question you will explore how the variability of mixed excitatory and inhibitory
synaptic inputs translates to the variability of the neural response.
Consider a
Chapter 1
The HodgkinHuxley Equations
1.1 The Resting Potential
All living cells have an electrical voltage, or potential difference, between their
inside and outside. Since the cells membrane is what separates the inside from
the outside, this potential
Chapter 11
Firing Rate Models
One of the most common ways to model large networks of neurons is to use a
simplication called a ring rate model. Rather than track the spiking of every neuron, instead one tracks the averaged behavior of the spike rates of g
Math 3375
MATH 1370 - Assignment 1
1. Synaptic integration and coincidence detection
Consider a leaky integrate-and-re neuron driven by a synaptic train. The membrane
dynamics of the neuron obeys:
dV
+ gCL (V VCL ) + gsyn y(t)(V Vsyn ),
dt
supplemented wi
MATH 1370 Project
Computational neuroscience is a eld that applies mathematical analysis to problems motivated by experimental ndings in neuroscience. However, there are many experimental
sub-communities within neuroscience, and naturally computational ne
Math 1370
MATH 1370 - Assignment 4
1. Firing rate adaptation and competition models. In this problem you will consider the eects of spike rate adaptation on ring dynamics. Particular attention will
be given to two population models which inhibit one anoth
Math 3375
MATH 1370 - Assignment 2
1. Class I and Class II excitability
Consider the planar neural model
dV
+ gN a m (V )(V VK ) + gK n(V VK ) + gCL (V VCL ) = I(t),
dt
dn
= n (V ) n.
n (V )
dt
C
Parameters and functions can be found in the MATLAB le Pres