Simple approaches using cellular parameters (eg, Rm, Cm, ~' GNa, etc.) can predict many effects of dendritic spiking, as confirmed by detailed compartmental simulations of the reconstructed pyramidal cell. The task of the population of neurons is to respond to multiple simultaneous stimuli while still identifying those neurons that respond to a particular stimulus.
Chapter 1
Introduction
- The Basic Questions
- INTROD UCTION 2
- The Average Rate Code
- INTRODUCTION 3
- Why Noise Isn't Investigated
- INTRODUCTION 4
- INTRODUCTION 5
- Why "Noise" Might Be Information
- INTRODUCTION 6
- INTRODUCTION 7
- INTRODUCTION 8
- Thesis Overview
- INTRODUCTION 9
- INTRODUCTION 10 carried by precise spike-times in most cortical areas. Only temporal averages
- INTRODUCTION 11
- Cortical Physiology Oversimplified
- INTRODUCTION 12
- INTRODUCTION 13
- INTRODUCTION 14
- INTRODUCTION
- Chapter 2
Some electrophysiologists have focused on the idea that the dynamics of the neuronal response may contain important information (McClurkin et al.). The final effect is that the strong depolarization of the cell causes the opening of potassium channels.
A Paradox: Cortical cells do not perform temporal
Introduction
A PARADOX 17
A PARADOX 18
- Electrophysiological Data
A PARADOX 19 recordings came from an investigation of the influence of the larger "non-
A PARADOX 20
A PARADOX 22
- Analysis Method
- Parameters and Normalization Procedure
A PARADOX 23 histograms with a narrow peak on a broad base, which would be excluded by
A PARADOX 24 separately for each cell, as follows
A PARADOX 25 Rmax (defined over all stimulus conditions for that cell) was used to define
The instantaneous speed R;(t) for train j was calculated by multiplying r(t) by the ratio of the total number of points S; of that train for the average number of spikes of cell Savg· Here, Savg = 83 and S; = 131. Each lSI was placed on one of ten lSI histograms, so that each histogram represented a constant readiness rate: histogram #0 was the slowest and histogram #9 was the fastest.
- Inaccuracy of Analysis Method
The main systematic bias of the analysis method was the underestimation of Cv for large lSIs. The slightly higher firing rate of Vl neurons resulted from the selection of such faster neurons for analysis; there are no other notable differences between the two areas.
A PARADOX 29
We analyzed these spurious trains with the same method used for the monkey data, analyzing slow and fast trains separately to resolve Cv at long and short lSI values. The assay method underestimated high Cv at long lSI values and overestimated low Cv at short lSI values.
A PARADOX 32
A PARADOX 33
- Variability in the Interspike Interval
- Variability in the Number of Spikes
A PARADOX 34
- Analytical Models
- Integrate-and-Fire Neuron
Plots of the number of spikes Si in a train for a continuous stimulus and the variance u~ in that number indicate the variability of firing over longer times; the log-log scale contains values from a few peaks to hundreds. Values for monkey cells are crosses, consistent with values obtained for the same areas by Snowden et al.
A PARADOX 36
A PARADOX 37
A PARADOX 38
- Refractory Period
A PARADOX 39
- Leaky Integrate-and-Fire Neuron
30 Kt(msec)
Our predictions for the Cv of this model come from numerical simulation of Eq. 2.15, using a real value for the membrane time constant of T= 13 msec. The conflict between theory and data is greatest for faster-firing cells (!:::.t < T ~ 13 msec); in that regime the flow integrator prediction is approx.
A PARADOX 42
- Realistic Parameters and Modifications
- msec 1.0 msec 10 msec
The lower curve shows the simulated leaky integrator model with parameter values in the accepted range (Nth = 51, r = 13 msec and t0 = 1.0 msec). The upper curve shows the theoretical upper limit of Cv for a pure Poisson spike train with "dead time" t0 = 1.0 msec.
A PARADOX 46
A PARADOX 47
- Compartment Models
A PARADOX 48
- Biophysical Modeling of a Cortical Pyramidal Cell
A PARADOX 49
A PARADOX 50
A PARADOX 51
A PARADOX 52
A PARADOX 53
Somatic tension in the "conventional" simulation, with. tpeak = 1.5 msec, 9mar = 0.5 nS, with excitatory synapses randomly distributed throughout the dendritic tree. right). On the right are histograms of the "conventional" model (shown here as analyzed using 0.1 msec bins and 20 histograms for greater resolution).
A PARADOX 59
A PARADOX 60
- Comparison of Compartmental and Analytical Model
A PARADOX 61 integrator with refractory period, modified for adaptation and random-height,
A PARADOX 62
All plots show simulations of the "conventional" model with fixed-amplitude synaptic currents (the "baseline" model, open squares) and predictions of the integrator model with refractory period 1.5 ms, Nde = 80, and peak-adaptation currents ( thin curves). Synaptic conductance values fivefold faster than the baseline model's (but with similar area) lead to a predicted increase in firing variability at the highest rate (thick curve); no such increase is evident (filled boxes).
A PARADOX 64 refractory period requires many more than Nth EPSPs, so that the output
- Active Dendritic Simulation
A PARADOX 65
A PARADOX 66
A PARADOX 67
Dendritic spikes (thin line in inset) were activated in each of 42 active basal branches at a random time more than 2 msec after its previous firing, and independently of the other branches' firing. The most variable somatic firing occurred for dendritic potassium conductances 9DR twice the strength of the sodium conductance, because the strong repolarization short-circuited the somatic depolarization (thick line in inset), thereby preventing temporal integration.
A PARADOX 69
- Discussion
A PARADOX 70
- Statistical Assumption Underlying our Data Anal-
A PARADOX 71 tens of milliseconds. In a different study;we had computed the autocorrelation
- The Variability of Cortical Cell Firing
A PARADOX 72
- Analytical Results
A PARADOX 73
- Biophysical Detailed Simulations
A PARADOX 74 rapidly firing interneurons rather than pyramidal cells ( Agmon and Connors
A PARADOX 75 Similar to the integrate-and-fire model discussed above, high Cv values could
A PARADOX 76
A PARADOX 77
- Network Effects
A PARADOX 78
A PARADOX 79
A PARADOX 80
- Conclusion
A PARADOX 81
Chapter 3
A Solution: submillisecond
- Introduction
- A SOLUTION 83
- A SOL UTI ON
- Cable Theory at Fast Timescales
- A SOL UTI ON 85
- A SOLUTION 86
- A SOLUTION 87
- A SOL UTI ON 88 voltage as the peak of the approximate distribution,
- A SOLUTION 89 branches of various diameters and geometries, this approximation shows the
- Simulated Pyramidal Cell
- A SOLUTION 90
- A SOLUTION 92
- Fast EPSPs in Thin Terminal Branches
- A SOLUTION 93
- A SOLUTION 94
- A SOL UTI ON 95 charge distributes itself over a capacitance C(t) on both sides of the synaptic
- A SOLUTION 97 This very narrow pulse-width matches almost exactly the t 1; 2 of EPSPs with
- A SOLUTION 98
- Active Dendritic Terminal Branches
- A SOLUTION 99
- A SOLUTION 100 eral strikingly simple approximations. First, we can suppose that instead of a
We must first ask which properties of the terminal branch itself will dominate: capacitive or resistive. However, the capacitance of the soma and other terminal branches together will be large enough that they will not become very depolarized during a single dendritic spike.
A SOLUTION 102
This input conductance assumes that the entire active portion of the dendrite contains uniform, half-open sodium channels, although it is clear that the most proximal channels are fully closed and the most distal are fully open. An approximation of the peak current comes from choosing a representative input conductance and voltage for that region and combining them into a current source whose magnitude depends only on membrane conductance and dendrite diameter (Section 3.4). B) When a terminal branch (black) is connected to a soma by a long dendritic trunk (grey), the full length of the branch can be exceeded, making the above current source approximation invalid.
A SOLUTION 105
- Somatic Depolarization from a Spike
- Coupling Between Active Terminal Branches
- Predicted Depolarization of Neighboring Branches
Let us first use the simplest possible approach: we assume that the peak current from B1 (eq. 3.41) is the dominant effect, and that it reaches equilibrium in the local strains, so that the depolarization at B2 is approximated by the depolarization at the distal end of T (Figure 3. 7 B) (this approach ignores the dendrites' capacitive and leakage effects). But if the stem voltage drop dominates (.6.VT ~ .6.Vaoma), then we take the peak in the dendrite as the stem voltage drop, added to the somatic voltage that exists at that time.
A SOLUTION 112
A SOL UTI ON 113 end of several representative terminal branches sharing part of the same stem. To calculate the magnitude of the depolarization at B2 due to B1, a simplified model assumes that B2 is directly connected to the same dendritic trunk as B1, that B1 provides peak current in accordance with eq. 3.40, and that the two terminal branches share a common portion of dendritic trunk T with resistance RT.
A SOLUTION 115
- Recruitment of Neighboring Branches by Spikes
VB2 plotted against the predicted depolarization (Figure 3.7 and section 3.5) has points close to the diagonal line representing perfect predictions: (A). 3.52 (points lying on the diagonal line represent a perfect prediction): (A) GNa = 0.033 S cm-2; (B) GNa = 0.2 S cm-2 • Points well above that line represent synchronous firing of one or more adjacent active dendrites activated by the firing of the first.
A SOLUTION 120
- Somatic Repolarization by Dendritic Spik-
- Pulse Widths
A dimensionless measure of sustained depolarization, .6.p, is the ratio of the somatic potential 8 ms after triggering (filled circles) to its value in the virtual absence of potassium currents (open circles); .6.p = 1 represents no attenuation due to potassium currents, and Ap < 0 represents a peak that leaves the cell more polarized than before. Only at moderately strong GK/GNa > 2 and moderately high Ere8t = -65 mV can dendritic ]DR currents remove persistent depolarization, a condition necessary for efficient coincidence detection between dendritic spikes.
A SOL UTI ON 123
- Capacitance of Dendritic Spines
For larger sodium conductances (GNa = 0.2 S cm-2), larger depolarizations sometimes recruited several branches to fire sequentially, extending the somatic pulse to values near 1.5 ms (rare cases in which all firing dendrites are not shown here). Depolarizations of such short duration can be used to perform submillisecond coincidence detection between dendritic spikes.
A SOLUTION 125
- Quantifying Coincidence-Detection
A SOLUTION 126
A SOL UTI ON 127
- Integrator-Models as Coincidence Detectors
But its response to optimally timed coincident inputs (in Nth waves) is the same as that of the perfect integrator model because the cell fires before it can be repolarized by the leak term:. For a fixed input rate rib, both leaky and fireproof models respond to coincident (or optimal) inputs as perfect integrators.
A SOLUTION 130 input rates. Thus at high rates coincident inputs are more effective than evenly
- Pyramidal Cells as Coincidence Detectors
A SOLUTION 131
A SOLUTION 133 evenly, a given synapse would participate in different groups on subsequent
This cell requires simultaneous EPSPs on the same branch (an AND function), but fires the soma when a branch fires (OR function), so it can be called an "AND-OR" cell. This cell performs very little temporal integration of dendritic spikes, so its effectiveness Ec as a coincidence detector is high (near or equal to unity); it approximates an “AND” operation of M dendritic spikes.
A SOLUTION 135
With strong dendritic conductances with a delayed rectifier (left column; GK = 0.066 Scm-2), these somatic voltage plots show that individual dendritic spikes rapidly repolarize the soma to about -65 mV so that their time integration does not occur; the cell will not fire in response to evenly spaced dendritic spikes, but only if they are coincident (A, B). Low values of Ec (F) indicate that this cell cannot distinguish between coincident and evenly spaced dendritic spikes.
A SOL UTI ON 137
- Discussion
- Requirements for Submillisecond Computation
A SOLUTION 139
A SOLUTION 140
A SOLUTION 141
- Plausibility and Testability of Critical Assump- tions tions
A SOLUTION 142
A SOLUTION 143 Active Dendritic Conductances
A SOL UTI ON 144
A SOLUTION 145
A SOLUTION 146 consistent with published cross-correlation data (Appendix D; the following
A SOLUTION 147
- Analytical and Simulation Results
A SOLUTION 148
- Conclusion
A SOLUTION 149 and visual awareness (Crick and Koch 1990). But it remains to be seen exper-
Chapter 4
An Application
Point-of-Origin Binding
Introduction
AN APPLICATION 151 there is no simple way to determine which active neuron refers to which object
AN APPLICATION 151 there is no simple way to determine which active neuron refers to which object. In this example, there is a chair (not moving), a moving cat, and a moving girl, all of which stimulate the neurons.
AN APPLICATION 153
AN APPLICATION 154
AN APPLICATION
- Cartoon Example
34;on,” and no pulses if it is “off.” The top neuron, with no input, produces no output; the bottom one has three active pixels and fires on their occasional coincidences.
AN APPLICATION 158
- Shape-Detector Neurons
Each neuron has a receptive field the size of the entire visual space and is designed to respond if its preferred shape appears in one of its subunits anywhere in that space. The sub-unit turns on at random every two points from the pixels in its input (in the same way as the location detector does).
AN APPLICATION 160
- Binding the Outputs
Different neurons in both classes will fire: the 'X' detector and the 'T' detector (both firing at rate Ps ), and various location detectors. One can show that a T will overlap an X detector and an X will overlap a T detector in six ways to drive two pixels (As = 2).
AN APPLICATION 162 Now we can try to reconstruct the inputs from the outputs. If one only looks
Note that of the pixels that drive the location detectors, in both cases three of those pixels also drive the optimal shape detector (inverted triangle in Figure 4.3). While the average firing rates of these neurons do not indicate which shape is at which location, correctly matched neuron pairs will share a higher-than-random rate of coincident output spikes as a result of their three common input sets of spikes, so that the original patterns can in principle be reconstructed.
AN APPLICATION 164
AN APPLICATION 165
- Biological Implausibilities
AN APPLICATION 166 for temporal input patterns, no orientation selectivity, no distinction between
- Special Features
AN APPLICATION 167 detection. There is also no reason that the driver neurons need to be individual
- Advantages
- Conclusion
Encoding explicitly requires the all-or-none nature of the action potential ("digital"), instead of using pulses to convey an analog firing rate. However complicated the world may be, its complexity pales in comparison to the complexity of the pattern space it inhabits.
AN APPLICATION 169
AN APPLICATION 170
AN APPLICATION 171
Appendix A
EPSP Width
EPSP WIDTH 173 some test frequency j; blurring is given by the reduction in the Fourier amplitude of the blurred current train at f relative to the unblurred spike train. Assuming that this attenuation of the current roughly corresponds to the attenuation of Cv (see Eq.
Appendix B
Spike Adaptation
Recognizing that the observed lSI contains the refractory period t0, this means that the true integration time is /:).t - to,. Indeed, such a sharp increase in variability is observed both in simulations (figures) and in the monkey data.
Appendix C
Irregular EPSP Magnitude
The spread around the average number of EPSPs is .j]V;h, so we expect the depolarization to consist of Nth± .,JN;;, EPSPs, each of magnitude v±av. By applying our assumption that the standard deviation of the EPSP amplitude is approximately the same as the mean (av = v; Mason et al. 1991) and combining equations.
Appendix D
Cross- Correlation Analysis
- Influences of Recording Method
The height of the off-peak cross-correlogram CB-A(f:::.tmax)d(f:::.t) is chosen as the basis for determining the peak area. A shorter chance window estimate would lead to even lower Cc values and smaller contributions to variability.
