Contact Info
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext. 32700
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
MC 6496
Wilten Nicola | Applied Math, University of Waterloo
Bifurcation Analysis of Large Networks of Neurons
The human brain contains on the order of a hundred billion neurons, each with several thousand synaptic connections. Computational neuroscience has successfully modeled both the individual neurons as various types of oscillators, in addition to the synaptic coupling between the neurons. However, employing the individual neuronal models as a large coupled network on the scale of the human brain would require massive computational and financial resources, and yet is the current undertaking of several research groups. Even if one were to successfully model such a complicated system of coupled differential equations, aside from brute force numerical simulations, little insight may be gained into how the human brain solves problems or performs tasks.
Here, we introduce a tool that reduces large networks of coupled neurons to a much smaller set of differential equations that governs key statistics for the network as a whole, as opposed to tracking the individual dynamics of neurons and their connections. This approach is typically referred to as a mean field system. As the mean field system is derived from the original network of neurons, it is predictive for the behavior of the network as a whole and the parameters or distributions of parameters that appear in the mean field system are identical to those of the original network. As such, bifurcation analysis is predictive for the behavior of the original network and predicts where in the parameter space the network transitions from one behavior to another.
Additionally, here we show how networks of neurons can be constructed with a mean field or macroscopic behavior that is prescribed. This occurs through an analytic extension of the Neural Engineering Framework (NEF). This can be thought of as an inverse mean field approach, where the networks are constructed to obey prescribed dynamics as opposed to deriving the macroscopic dynamics from an underlying network. Thus, the work done here analyzes neuronal networks through both top-down and bottom-up approaches.
S | M | T | W | T | F | S |
---|---|---|---|---|---|---|
29
|
30
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
25
|
26
|
27
|
28
|
29
|
30
|
31
|
1
|
2
|
Contact Info
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext. 32700
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is co-ordinated within the Office of Indigenous Relations.