Spikes, Decisions, ActionsThe dynamical foundations of neuroscience

Valance WANG

Computational Biology and Bioinformatics, ETH Zurich

The last meeting• Higher-dimensional linear dynamical systems

• General solution• Asymptotic stability• Oscillation• Delayed feedback

• Approximation and simulation

Outline• Chapter 6. Nonlinear dynamics and bifurcations

• Two-neuron networks• Negative feedback: a divisive gain control• Positive feedback: a short term memory circuit• Mutual Inhibition: a winner-take-all network

• Stability of steady states• Hysteresis and Bifurcation

• Chapter 7. Computation by excitatory and inhibitory networks• Visual search by winner-take-all network• Short term memory by Wilson-Cowan cortical dynamics

Chapter 6. Two-neuron networks Input Input

Input Input

Two-neuron networks• General form (in absence of stimulus input):

• Reading current state as input to the update function • Steady states:

Negative feedback: a divisive gain control

• In retina,• Light -> Photo-receptors -> Bipolar cells -> Ganglion cells -> optic

nerves• Amacrine cell

• This forms a relay chain of information• To stabilize representation of information, bipolar cells receive

negative feedback from amacrine cell

Negative feedback: a divisive gain control

• In retina,

Negative feedback: a divisive gain control

• Equations:B A

Light

• Equations:• Nullclines:

• Equilibrium point:

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phase plane analysis for L=10

dB/dt=0dA/dt=0

Linear stability of steady states• Introduction to Jacobian:• Given • Jacobian

• Example: given our update function

• Jacobian

Linear stability of steady states

Linear stability of steady states• Proof:• Our equations

• Apply a small perturbation to the steady state, u,v << 1, take this point as initial condition

• Where , u(t),v(t) represents deviation from steady states

• Proof (cont.):• Plug in and solve

• Finally•

• Then use eigenvalue to determine asymptotic behavior

Negative feedback: a divisive gain control

• Equations:• Fixed point • Stability analysis

• Jacobian at (2,4) =

• Eigenvalues => asymptotically stable• Unique stable fixed point => our fixed point is a «global attractor»

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Two-neuron networks Input Input

Input Input

A short-term memory circuit by positive feedback

• In monkeys’ prefrontal cortex

A short-term memory circuit by positive feedback

• First, let’s analyze the behavior of the system in absence of external stimulus

• Equations:

E1 E2

• A sigmoidal activation function: • P: stimulus strength• S: firing rate

A short-term memory circuit by positive feedback

• Equations:

• Nullclines:

• Equilibrium point:

• E2eq can be obtained similarly

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phase plane analysis

dE1/dt=0dE2/dt=0

• Equilibrium point:

• Stability analysis:• (0,0): Jacobian • (20,20): Jacobian • (100,100): Jacobian

Hysteresis and Bifurcation• The term ‘hysteresis’ is derived from Greek, meaning ‘to lag

behind’.• In present context, this means that the present state of our

neural network is determined not just by the present state and input, but also by the state and input in the history (“path-dependent”).

Hysteresis and Bifurcation• Suppose we apply a brief stimulus K to the neural network

• The steady states of E1 becomes

• Demo

E1 E2

K

Hysteresis and Bifurcation• Due to change in parameter value K, a pair of equilibrium

points may appear or disappear. This phenomenon is known as bifurcation.

Two-neuron networks Input Input

Input Input

Mutual inhibition: a winner-take-all neural network for decision making

• Demo

K1

E1 E2

K2

Chapter 6. Two-neuron networks Input Input

Input Input

Chapter 7. Multiple-Neuron-network

• Visual search by a winner-take-all network• Wilson-Cowan cortical dynamics

Visual search by winner-take-all network

• Visual search

Visual search by winner-take-all network

• A N+1 Neuron-network, each neuron receives perceptive input• T for target, D for distractor

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• Stimulus to target neuron:80, to disturbing neurons:79.8

• Stimulus to target neuron: 80, to disturbing neurons: 79

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winner neuron

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winner neuron

• Further, this model can be extrapolated for higher level cognitive decisions. It is common experience that decisions are more difficult to make and take longer when the number of appealing alternatives increases.

• Once a decision is definitely made, however, humans are reluctant to change their decision. (Hysteresis in cognitive process!)

Wilson-Cowan model (1973)• Cortical neurons may be divided into two classes:

• excitatory (E), usu. Pyramidal neurons• and inhibitory (I), usu. interneurons

• All forms of interaction occur between these classes: • E -> E, E -> I, I -> E, I -> I

• Recurrent excitatory network are local, while inhibitory connections are long range

• A one-dimensional spatial-temporal model

• E(x,t), I(x,t) := mean firing rates of neurons • x := position • P,Q := external inputs• wEE, wIE, wEI, wII, := weights of interactions

• Spatial exponential decay is determined by, e.g.

• x := position of input• x’ := position away from the input

• Sigmoidal activation function

• P := stimulus input• Sigmoidal curve with respect to P

• Example: short term memory in prefrontal cortex• A brief stimulus = 10ms, 100 µm

• A brief stimulus = 10ms, 1000 µm

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Wilson-Cowan model• Examples: short term memory, constant stimulus

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Summary of Chapter 7• Winner-take all network

• Visual search can be disturbed by the number of irrelevant but similar objects

• Wilson-Cowan model• A one-dimensional spatial-temporal dynamical system

• Applications:• Short term memory in prefrontal cortex