Activity Dependent Conductances: An “Emergent” Separation of Time-Scales David McLaughlin Courant Institute & Center for Neural Science New York University

Input Layer of Primary Visual Cortex (V1) for Macaque Monkey Modeled at : Courant Institute of Math. Sciences & Center for Neural Science, NYU In collaboration with: Robert Shapley (Neural Sci) Michael Shelley Louis Tao Jacob Wielaard

Visual Pathway: Retina --> LGN --> V1 --> Beyond

Our Model A detailed, fine scale model of a local patch of input layer of Primary Visual Cortex; Realistically constrained by experimental data; Refs: McLaughlin, Shapley, Shelley & Wielaard --- PNAS (July 2000) --- J Neural Science (July 2001) --- J Neural Science (submitted, 2001) Today 

Equations of the Model  = E,I v j  -- membrane potential --  = Exc, Inhib -- j = 2 dim label of location on cortical layer neurons per sq mm (12000 Exc, 4000 Inh) V E & V I -- Exc & Inh Reversal Potentials (Constants)

Conductance Based Model  = E,I Schematic of Conductances g  E (t) = g LGN (t) + g noise (t) + g cortical (t)

Conductance Based Model  = E,I Schematic of Conductances g  E (t) = g LGN (t) + g noise (t) + g cortical (t) (driving term)

Conductance Based Model  = E,I Schematic of Conductances g  E (t) = g LGN (t) + g noise (t) + g cortical (t) (driving term) (synaptic noise) (synaptic time scale)

Conductance Based Model  = E,I Schematic of Conductances g  E (t) = g LGN (t) + g noise (t) + g cortical (t) (driving term) (synaptic noise) (cortico-cortical) (synaptic time scale) (L Exc > L Inh ) (Isotropic)

Conductance Based Model  = E,I Schematic of Conductances g  E (t) = g LGN (t) + g noise (t) + g cortical (t) (driving term) (synaptic noise) (cortico-cortical) (synaptic time scale) (L Exc > L Inh ) (Isotropic) Inhibitory Conductances: g  I (t) = g noise (t) + g cortical (t)

Integrate & Fire Model  = E,I Spike Times: t j k = k th spike time of j th neuron Defined by: v j  (t = t j k ) = 1, v j  (t = t j k +  ) = 0

Conductances from Spiking Neurons LGN & Noise Spatial Temporal Cortico-cortical  Here t k l (T k l ) denote the l th spike time of k th neuron

Elementary Feature Detectors Individual neurons in V1 respond preferentially to elementary features of the visual scene (color, direction of motion, speed of motion, spatial wave-length).

Elementary Feature Detectors Individual neurons in V1 respond preferentially to elementary features of the visual scene (color, direction of motion, speed of motion, spatial wave-length). Three important features:

Elementary Feature Detectors Individual neurons in V1 respond preferentially to elementary features of the visual scene (color, direction of motion, speed of motion, spatial wave-length). Three important features: Spatial location (receptive field of the neuron)

Elementary Feature Detectors Individual neurons in V1 respond preferentially to elementary features of the visual scene (color, direction of motion, speed of motion, spatial wave-length). Three important features: Spatial location (receptive field of the neuron) Spatial phase  (relative to receptive field center)

Elementary Feature Detectors Individual neurons in V1 respond preferentially to elementary features of the visual scene (color, direction of motion, speed of motion, spatial wave-length). Three important features: Spatial location (receptive field of the neuron) Spatial phase  (relative to receptive field center) Orientation  of edges.

Two Angles: Angle of orientation --  Angle of spatial phase --  (relevant for standing gratings) Grating Stimuli Standing & Drifting

Orientation Tuning Curves (Firing Rates Vs Angle of Orientation) Terminology: Orientation Preference Orientation Selectivity Measured by “ Half-Widths” or “Peak-to-Trough” Spikes/sec 

Orientation Preference

Model neurons receive their orientation preference from convergent LGN input;

Orientation Preference Model neurons receive their orientation preference from convergent LGN input; How does the orientation preference  k of the k th cortical neuron depend upon the neuron’s location k = (k 1, k 2 ) in the cortical layer?

Cortical Map of Orientation Preference Optical Imaging Blasdel, 1992 Outer layers (2/3) of V1 Color coded for angle of orientation preference ----  500   ----  right eye  left eye

Pinwheel Centers

4 Pinwheel Centers 1 mm x 1 mm

Active Model Cortex - High Conductances When the model performs realistically, with respect to biological measurements – with proper -- firing rates -- orientation selectivity (tuning width & diversity) -- linearity of simple cells the numerical cortex resides in a state of high conductance, with inhibitory conductances dominant! The next few slides demonstrate this “cortical operating point” \

Conductances Vs Time Drifting Gratings -- 8 Hz Turned on at t = 1.0 sec Cortico-cortical excitation weak relative to LGN; inhibition >> excitation

Distribution of Conductance Within the Layer = Time Average  SD(g T ) = Standard Deviation Of Temporal Fluctuations  Sec -1

Active Model Cortex - High Conductances Background Firing Statistics ====> g Back = 2-3 g slice

Active Model Cortex - High Conductances Background Firing Statistics ====> g Back = 2-3 g slice Active operating point ====> g Act = 2-3 g Back = 4-9 g slice

Active Model Cortex - High Conductances Background Firing Statistics ====> g Back = 2-3 g slice Active operating point ====> g Act = 2-3 g Back = 4-9 g slice ====> g Inh >> g Exc

Active Model Cortex - High Conductances Background Firing Statistics ====> g Back = 2-3 g slice Active operating point ====> g Act = 2-3 g Back = 4-9 g slice ====> g Inh >> g Exc Consistent with experiment Hirsch, et al, J. Neural Sci ‘98; Borg-Graham, et al, Nature ‘98; Anderson, et al, J. Physiology ‘00

Active Cortex - Consequences of High Conductances Separation of time scales ;

Active Cortex - Consequences of High Conductances Separation of time scales ; Activity induced  g = g T -1 <<  syn (actually, 2 ms << 4 ms)

Conductance Based Model  = E,I dv/dt = - g T (t) [ v - V Eff (t) ], where g T (t) denotes the total conductance, and V Eff (t) = [ V E g EE (t) - | V I | g EI (t) ] [ g T (t) ] -1 If [g T (t)] -1 <<  syn  v  V Eff (t)

But the separation is only a factor of 2 (  g = g T -1 = 2 ms;  syn =4 ms) Is this enough for the time scales to be “well separated” ?

Active Cortex - Consequences of High Conductances Membrane potential ``instantaneously’’ tracks conductances on the synaptic time scale. V(t) ~ V Eff (t) = [ V E g EE (t) - | V I | g EI (t) ] [ g T (t) ] -1 where g T (t) denotes the total conductance

High Conductances in Active Cortex  Membrane Potential Tracks Instantaneously “Effective Reversal Potential” Active Background

Effects of Scale Separation  g = 2  syn  g =  syn  g = ½  syn ____(Red) = V Eff (t) ____(Green) = V(t)

Fluctuation-driven spiking Solid: average ( over 72 cycles) Dashed: 10 temporal trajectories (very noisy dynamics, on the synaptic time scale)

Coarse-Grained Asymptotics

Using the spatial regularity of cortical maps (such as orientation preference), we “coarse grain” the cortical layer into local cells or “placquets”.

Cortical Map of Orientation Preference Optical Imaging Blasdel, 1992 Outer layers (2/3) of V1 Color coded for angle of orientation preference ----  500   ----  right eye  left eye

Coarse-Grained Asymptotics Using the spatial regularity of cortical maps (such as orientation preference), we “coarse grain” the cortical layer into local cells or “placquets”.

Coarse-Grained Asymptotics Using the spatial regularity of cortical maps (such as orientation preference), we “coarse grain” the cortical layer into local cells or “placquets”. Using the separation of time scales which emerge from cortical activity,  g <<  syn

Coarse-Grained Asymptotics Using the spatial regularity of cortical maps (such as orientation preference), we “coarse grain” the cortical layer into local cells or “placquets”. Using the separation of time scales which emerge from cortical activity,  g <<  syn

Coarse-Grained Asymptotics Using the spatial regularity of cortical maps (such as orientation preference), we “coarse grain” the cortical layer into local cells or “placquets”. Using the separation of time scales which emerge from cortical activity,  g <<  syn Together with an averaging over the random cortical maps (such as spatial phase, De Angelis, et al ‘99)

Coarse-Grained Asymptotics Using the spatial regularity of cortical maps (such as orientation preference), we “coarse grain” the cortical layer into local cells or “placquets”. Using the separation of time scales which emerge from cortical activity,  g <<  syn Together with an averaging over the irregular cortical maps (such as spatial phase) we derive a coarse-grained description in terms of the average firing rates of neurons within each placquet [--- a form of Cowan – Wilson Eqs (1973)]



Uses of Coarse-Grained Eqs

Unveil mechanims for (i) Better orientation selectivity near pinwheel centers (ii) Balances for simple and complex cells Input-output relations at high conductance Comparison of the mechanisms and performance of distinct models of the cortex Most importantly, much faster to integrate; Therefore, potential parameterizations for more global descriptions of the cortex.

Active Cortex - Consequences of High Conductances Cortical activity induces a separation of time scales (with the synaptic time scale no longer the shortest), Thus, cortical activity can convert neurons from integrators to burst generators & coincidence detectors.  For transmission of information: “Input” temporal resolution -- synaptic time scale  syn ; “Output” temporal resolution --  g = g T -1

Summary: One Model of Local Patch of V1 A detailed fine scale model -- constrained in its construction and performance by experimental data ; Orientation selectivity & its diversity from cortico-cortical activity, with neurons more selective near pinwheels; Linearity of Simple Cells -- produced by (i) averages over spatial phase, together with cortico-cortical overbalance for inhibition; Complex Cells -- produced by weaker (and varied) LGN input, together with stronger cortical excitation; Operates in a high conductance state -- which results from cortical activity, is consistent with experiment, and makes integration times shorter than synaptic times, a separation of temporal scales with functional implications; Together with a coarse-grained asymptotic reduction -- which unveils cortical mechanisms, and will be used to parameterize or ``scale-up’’ to larger more global cortical models.

Scale-up & Dynamical Issues for Cortical Modeling Temporal emergence of visual perception Role of temporal feedback -- within and between cortical layers and regions Synchrony & asynchrony Presence (or absence) and role of oscillations Spike-timing vs firing rate codes Very noisy, fluctuation driven system Emergence of an activity dependent, separation of time scales But often no (or little) temporal scale separation

Distribution of Conductances Over Sub-Populations “FAR” & “NEAR” Pinwheel Centers = Time Average  SD(g T ) = Stand Dev of Temporal Fluctuations 

One application of Coarse-Grained Equations

Why the Primary Visual Cortex?

Elementary processing, early in visual pathway Neurons in V1 detect elementary features of the visual scene, such as spatial frequency, direction, & orientation

Why the Primary Visual Cortex? Elementary processing, early in visual pathway Neurons in V1 detect elementary features of the visual scene, such as spatial frequency, direction, & orientation Vast amount of experimental information about V1

Why the Primary Visual Cortex? Elementary processing, early in visual pathway Neurons in V1 detect elementary features of the visual scene, such as spatial frequency, direction, & orientation Vast amount of experimental information about V1 Input from LGN well understood (Shapley, Reid, …) Anatomy of V1 well understood (Lund, Callaway,...)

Why the Primary Visual Cortex? Elementary processing, early in visual pathway Neurons in V1 detect elementary features of the visual scene, such as spatial frequency, direction, & orientation Vast amount of experimental information about V1 Input from LGN well understood (Shapley, Reid, …) Anatomy of V1 well understood (Lund, Callaway,...) The cortical region with finest spatial resolution --

Why the Primary Visual Cortex? Elementary processing, early in visual pathway Neurons in V1 detect elementary features of the visual scene, such as spatial frequency, direction, & orientation Vast amount of experimental information about V1 Input from LGN well understood (Shapley, Reid, …) Anatomy of V1 well understood (Lund, Callaway,...) The cortical region with finest spatial resolution -- Detailed visual features of input signal;

Why the Primary Visual Cortex? Elementary processing, early in visual pathway Neurons in V1 detect elementary features of the visual scene, such as spatial frequency, direction, & orientation Vast amount of experimental information about V1 Input from LGN well understood (Shapley, Reid, …) Anatomy of V1 well understood (Lund, Callaway,...) The cortical region with finest spatial resolution -- Detailed visual features of input signal; Fine scale resolution available for possible representation;