A Catastrophe Theory Model for Cognition Decline
The Cognitive Network As Dependent on Metabolic Activity and
Synaptic Structure: A Trip through Cognitive Decline to Dementia
Oscar Sotolongo-Costa 1,2 L. M. Gaggero-Sager 1
J. T. Becker 4 O. Sotolongo-Grau 5
1 Departamento
2 ”Henri
de Física, Facultad de Ciencias. Universidad Autónoma del Estado de
Morelos. México
Poincaré” Group of Complex Systems, University of Havana, Cuba
3 Center
4 Department
F. Maestu 3
for Biomedical Technology, Spain
of Psychology, Department of Neurology, Department of Psychiatry, School of
Medicine, University of Pittsburgh, USA
5 Fundació
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ACE. Institut Català de Neurociències Aplicades. Spain
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Objective
Objective
• Find a simple mathematical model that describe the cognitive
decline through dementia
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Introduction
The cognition network
• Well, first let us consider a spherical cow brain ...
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Introduction
The cognition network
• Well, first let us consider a spherical cow brain ...
Actually, this is just a topological problem
=⇒
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Introduction
The cognition network
The neurons could be seen as the nodes of a network over this
surface. The links form the 3D body of the network.
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Introduction
The cognition network
A cognition network is the network that is used to develop a specific
cognitive function.
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Introduction
The first law of thermodynamics
• For a closed system, in any process, the change in the internal
energy is considered due to a combination of heat added to the
system and work done by the system
• For the mean cognition network, in any cognition process, the
change in metabolic activity is considered due to a combination of
the metabolic losses and the cognition developed by the network
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Introduction
The first law of thermodynamics
• For a closed system, in any process, the change in the internal
energy is considered due to a combination of heat added to the
system and work done by the system
• For the mean cognition network, in any cognition process, the
change in metabolic activity is considered due to a combination of
the metabolic losses and the cognition developed by the network
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The model of cognition
Spending the energy
• The cognition is proportional to the correlation length of the
network (y )
• The metabolic activity (m) developed by the network is
proportional to the network volume
• There is an amount of metabolic energy that is lost
m = m0 + ay 3
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The model of cognition
Spending the energy
• The cognition is proportional to the correlation length of the
network (y )
• The metabolic activity (m) developed by the network is
proportional to the network volume
• There is an amount of metabolic energy that is lost
m = m0 + ay 3
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The model of cognition
Saving the energy
Brain network is a trade off between energetic cost and efficiency
Lattice
cheap,
inefficient
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Complex
trade off
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expensive,
efficient
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The model of cognition
Saving the energy
• The network is organized as a complex network, gaining efficiency
and spending the energy as a smaller network of length y − y0
m = m0 + ay 3 −→ m = m0 + a (y − y0 )3
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The model of cognition
Saving the energy
• The network is organized as a complex network, gaining efficiency
and spending the energy as a smaller network of length y − y0
m = m0 + ay 3 −→ m = m0 + a (y − y0 )3
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The model of cognition
Saving the energy
Several networks maintain the same nodes
Cognition network A
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Cognition network B
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The model of cognition
Saving the energy
Overlaping nodes also save energy
Networks overlap in the red marked nodes
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The model of cognition
Saving the energy
• Some nodes do not need to be fully maintained by a single
cognition network
• To characterize this phenomenon the synaptic overlap x is defined
as a measure of the size of cortical connectivity hubs of the brain
• The amount of saved energy depends then on the volume of the
network that correspond to the shared cortical area
m = a (y − y0 )3 + m0 −byx 2
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The model of cognition
Saving the energy
• Some nodes do not need to be fully maintained by a single
cognition network
• To characterize this phenomenon the synaptic overlap x is defined
as a measure of the size of cortical connectivity hubs of the brain
• The amount of saved energy depends then on the volume of the
network that correspond to the shared cortical area
m = a (y − y0 )3 + m0 −byx 2
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Catastrophe theory model of cognition decline
The model of cognition
Cognition
m = m0 + a (y − y0 )3 − byx 2
Synaptic
overlap
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Metabolic activity
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Catastrophe theory model of cognition decline
The model of cognition
An important remark
• The cognition is described only with the metabolic activity and
synaptic overlap but those are not necessarily the ultimate
causes, they are just useful parameters to describe the process
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Catastrophe theory model of cognition decline
The cognition as function of metabolic activity
healthy
critical point
catastrophic fall
Cognition
impairment
maximum recovery
after fall
dementia
Metabolic activity
Cognition decline: As the metabolic activity decrease the cognition also
decays
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Catastrophe theory model of cognition decline
The cognition as function of metabolic activity
healthy
critical point
catastrophic fall
Cognition
impairment
maximum recovery
after fall
dementia
Metabolic activity
At some critical value the cognition outcome of the network suddenly drops to
very lower values, when this happens it is said that a catastrophe occurs
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Catastrophe theory model of cognition decline
The cognition as function of metabolic activity
20
18
16
synaptic
overlap
14
12
10
8
6
4
2
0
40
60
80
100
120
140
The critical fall point and the maximum level of recovery after the catastrophic
fall depends on the synaptic overlap
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Conclusions
Conclusions
• A very simple mathematical model for cognition decline has been
built. Only two (measurable!) quantities are need in order to
describe the cognition decline process.
• Beyond its simplicity the model is capable of predict very complex
phenomena.
• At some point of the cognitive decline a catastrophic fall into
dementia occurs
• The possible therapies must be applied before this happens or the
subject recovery will be compromised
• The model predictions are compatible with known phenomena as
compensation or brain reorganization
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Acknowledgments
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