Classical connectionist approach to computational neuroscience
Transcript of Classical connectionist approach to computational neuroscience
Spikes, Decisions, ActionsThe dynamical foundations of neuroscience
Valance WANGComputational 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
Nagative feedback
Positive feedback
Mutual inhibition
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
• Equations:
• Nullclines:
• Equilibrium point:
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B - bipolar cell response
A - amacrine cell response
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• 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
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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B - bipolar cell response
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phase plane analysis for L=10
dB/dt=0dA/dt=0
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 short-term memory circuit by positive feedback• Equations:
• Nullclines:
• Equilibrium point:
• E2eq can be obtained similarly
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E1
E2phase plane analysis
dE1/dt=0dE2/dt=0
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.
Chapter 6. Two-neuron networks
Nagative feedback
Positive feedback
Mutual inhibition
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• A N+1 Neuron-network, each neuron receives perceptive input
• T for target, D for distractorET
T D
ED
D
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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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• 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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E (red) & I (blue
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Wilson-Cowan model• Examples: short term memory, constant stimulus
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