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How Neurons Do Integrals. Mark Goldman. Outline. What is the neural basis of short-term memory? A model system: the Oculomotor Neural Integrator Neural mechanisms of integration: Linear network theory [4. Next lecture: Model critique – robustness in integrators and other neural systems].

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Presentation Transcript
slide2

Outline

  • What is the neural basis of short-term memory?
  • A model system: the Oculomotor Neural Integrator
  • Neural mechanisms of integration: Linear network theory
  • [4. Next lecture: Model critique – robustness in integrators and other neural systems]
slide3

input

∫input = running total

12

10

8

3

Add up the following numbers:

=10

3

+5

+4

-2

This task involves: 1. Addition

2. Memory

slide4

Record

voltage in

prefrontal

cortex

Voltage

Voltage

“spike”

Firing rate

(spikes/sec)

time

time

Sample

Memory period

(delay)

Match

Persistent neural activity is the neural correlate of short-term memory

characteristics of persistent neural activity
Characteristics of Persistent Neural Activity
  • Sustained elevated (or reduced) firing rates lasting up to ~10’s of seconds, long outlasting the stimulus
  • Rapid onset, offset
  • Mean firing rate an analog variable – a neuron can sustain any of many different levels of activity
  • Stable pattern of activity among many neurons
slide6

Model system:eye fixation

during spontaneous

horizontal

eye movements

Carassius auratus (goldfish)

the oculomotor neural integrator

time

Integrator

neurons

The Oculomotor Neural Integrator

E

Lateral (+)

Medial

Lateral

Eye

position E

(degrees)

Medial (-)

Motor neurons

(position signal)

readout

(+) direction

Eye movement

“burst” neurons

(velocity signal)

(-) direction

Other side neurons control opposite half of oculomotor range

slide8

persistent neural activity

(memory of eye position)

On/off

burst

input

+

+

+

Neural Recording from the Oculomotor Integrator

slide10

Many-neuron Patterns of Activity Represent Eye Position

saccade

Activity of 2 neurons

eye position represented

by location along

a low dimensional

manifold (“line attractor”)

(H.S. Seung, D. Lee)

slide11

General Principle?

Neural integrators operate by using

their input to shift internally-generated

patterns of activity along a coordinate

(coding) axis.

Persistent activity patterns are the

integrator “memory” of previous inputs

Activity Pattern= point on a line

line attractor picture of the neural integrator
Line Attractor Picture of the Neural Integrator

Geometrical picture

of eigenvectors:

Axes = firing rates

of 2 neurons

r2

r1

Decay along direction

of decaying eigenvectors

No decay or growth along direction

of eigenvector with eigenvalue = 1

“Line Attractor” or “Line of Fixed Points”

slide14

Effect of Bilateral Network Lesion

Bilateral Lidocaine:

Remove Positive Feedback

Control

unstable integrator
Unstable Integrator

Human with unstable integrator:

E

slide16

Weakness: Robustness to Perturbations

Imprecision in accuracy of feedback connections severely

compromises performance (memory drifts: leaky or unstable)

10% decrease in synaptic feedback

learning to integrate

Image

slip =

Eye slip

Need to turn up feedback

Learning to Integrate

How accomplish fine tuning of synaptic weights?

IDEA: Synaptic weights w learned from “image slip”

(Arnold & Robinson, 1992)

E.g. leaky integrator:

w

experiment give feedback as if eye is leaky or unstable
Experiment: Give Feedback as if Eye is Leaky or Unstable

Compute

image slip -

that would result

if eye were

leaky/unstable

=

E

E(t)

Magnetic coil

measures

eye position

integrator learns to compensate for leak instability
Integrator Learns to Compensate for Leak/Instability!

Control (in dark):

Give feedback

as if unstable

Leaky:

Give feedback

as if leaky

Unstable:

slide21

Summary

Persistent neural activity: activity which outlasts a transient stimulus

Neural integrator: outputs the mathematical integral of its inputs

-in the absence of input, maintains persistent neural activity

Oculomotor neural integrator:

-converts eye-velocity encoding inputs to eye position outputs

-Mechanism:

-Anatomy and physiology suggest network contribution

-Issue of Robustness:

Simple models are less robust to perturbations than the

real system

-Plasticity may keep the system tuned on slow time scales

-On faster time scales, intrinsic cellular processes may

provide a “friction-like” slowing of network decay

But: