Correction for Multi-Unit Clusters

Because spike amplitude is one of the main criteria for clustering neural units, two neurons which produce EAPs of similar amplitude at the electrode location will normally be clustered together. Such clusters are known as multi-unit clusters. Using my knowledge of the average range at which a neuron in each layer produces a spike of any given amplitude, I construct a model for the probability of recording multi-unit clusters. I can then correct the number of neurons recorded for the purpose of estimating the active cell fraction. I begin by describing an iterative procedure to achieve this, and then show that the result of the iterative procedure is equivalent to a convergent series with a simple analytic solution.

For two spikes to be clustered together, the amplitudes must be within a confined voltage range. I observed that my clustering algorithm ([Quiroga et al., 2004], and see section 4.2.2.2 below) would typically produce separate clusters for spikes with average amplitude that differed by at least 20µV. This value was relatively independent of the precise background noise level which mostly influence the minimum amplitude for spike clustering. Therefore I make the simple assumption that spikes that are less than around 20µV apart would be clustered together.

In order to simplify the analysis, I divide all spikes into ranges of 5µV rather than considering every spike individually. Actual clusters have amplitudes which vary from spike to spike, as discussed

in more detail in section 5.2.8, so little accuracy if any is lost here. I assumed that spikes that are within 10µV above and below each 5µV range would be clustered together. For example, a spike whose mean amplitude is between 90 and 95 µV would most likely be clustered together with any any other spikes which had mean amplitude in the range of 80-105µV that were present at the same location. So for convenience and to err on the side of caution I define 25µV as the range of amplitudes that would be clustered together, rather than 20µV. I will also analyze the impact if the multi-unit amplitude range were larger in section 4.3.7.

Assuming that the spike amplitude isosurfaces around the neurons are spherical, the probability that a given spike cluster is multi-unit depends on the intersection volume of two pairs of concentric spherical shells, as illustrated in Figure 4.2: the outer of each pair of shells is defined by the minimum voltage for the range, e.g., 80 µV. The inner of the two shells is defined by the maximum spike amplitude for the range, e.g., 105µV. That is, for a spike to be part of a multi-unit cluster it must be within the same amplitude range of two neighboring neurons, and they both must have been active during the recording. Therefore the (average) probability that a clustered spike in a given amplitude range is multi-unit is the probability that the spike was recorded in a position where a neighboring neuron would have an amplitude in the same range multiplied by the probability that the other neuron was in fact active at the same time. That is:

Pmulti=

Voverlap Vdetect

αNneighbors (4.2)

whereVoverlapis the volume of overlap for the detection regions of any single neighboring neuron

in the given voltage range, Vdetect is the total volume of the voltage range detection region for a

single neuron, αis the average probability that a neuron in the region is active, and Nneighbors is

the number of neighboring neurons at the same distance. This is not actually a probability because

Pmulti can be greater than one ifNneighbors is large enough – it is better thought of as a ratio, or

proportion. When I actually calculate the multi-unit corrections I assume the neurons lie on a cubic lattice so that each neuron has 6 nearest neighbors, 12 near diagonal neighbors (distance of√2 times the lattice size) and 8 far diagonal neighbors (distance of√3 times the lattice size).

The radii of the two outer (inner) shells for the two neighboring neurons are assumed to be the same, because the neurons under consideration would always be from the same layer and therefore have the same average detection ranges. Vdetectis calculated as the difference of the volume for the

two detection range shells. The next problem is to calculateVoverlap, the volume of the intersection

region. As illustrated in Figure 4.2, this is equivalent to:

Voverlap =Voo−2Voi+Vii (4.3)

and the inner shells, andVii is the volume of overlap for the two inner shells (Figure 4.2).

Voo

Vii

Voi

V_over

Figure 4.2: Illustration of multi-unit correction calculations.

I determine the intersection volume contained within two pairs of concentric shells, V. This volume is equivalent to the volume of overlap for the outer shells,Voo minus two times the volume of the

overlap of the outer shell with the inner shell,Voi, plus the volume of the overlap of the inner

shells,Vii.

The volume of the overlapping region for two neighboring spheres of unequal size was calculated in the study of hydration shells of molecules [Kang et al., 1987] which gives the following formula:

Vst= π 3(2r 3 s+ 2r 3 t+r 3 st)−π[r 2 sr ∗ st+ (rst−rst∗)(r 2 t+rsr∗st)] (4.4)

where rs andrtare the radii of the spheressand trespectively, rst is the distance between the

centers of the two spheres, andr∗_st is defined as:

r_st∗ = (r

2

s−r2t+r2st)

2rst

The final piece of information necessary for equation 4.2 is the fraction of neurons that are active. Unfortunately, this is what I am trying to determine. I propose the following iterative procedure to determine the fraction of active units corrected for multi-unit clusters in any given layer of cortex:

1. Calculate the multi-unit overlap fractionVoverlap/Vdetectfor each amplitude range, using equa-

tion 4.3 and 4.4.

2. Divide the recorded units into amplitude ranges

4. Update the number recorded in each range by multiplying the actual number recorded by the multi-unit recording probability, Pmulti, equation 4.2, using the initial value for αto yield a

new value for the total recorded, Nrecorded

5. Calculate an updated estimate forαusing equation 4.1 and the corrected value for Nrecorded

6. Repeat steps 4-5 until the value for αconverges.

Two objections to the proposed procedure could be: 1) that the result may depend on the initial guess for α; and 2) that it is not guaranteed to converge. In fact, the procedure described is equivalent to a simple non-homogeneous, linear recurrent sequence for the activity fraction that converges for all the practical situations I will encounter, no matter what initial guess is used for the activity fraction.

The iterative procedure described above results in a recurrent calculation for the activity fraction.

αn= (Rn/L)/U= Rn

LU (4.5)

whereαnis the estimated active neuron fraction on thenth iteration of the correction procedure, Rn is the (corrected) number of units recorded on the nth iteration, L is the number of recording

locations, andU is the number of units in electrode range predicted by the model. The update for the number of units recorded,Rn is:

Rn+1 = X s OsNsαn+N (4.6) = αnΩ +N (4.7) where Os= Voverlap

Vdetect Nneighbors is the the total shell overlap proportion for the amplitude range

indexed by s (Pmulti withoutα), Ns is the number of units recorded in the sth amplitude range,

andN is the total number of units recorded (P

sNs=N, but note thatOswill go to zero for higher

amplitude range.). For a compact notation I have defined Ω =P

sNsOs. Combining equations 4.5

and equations 4.6 there is a recurrence relationship forαn:

αn+1=

αnΩ +N

LU (4.8)

which is in the form of the simplest linear, non-homogeneous recurrent sequence:

xn=axn−1+b

xn=anx0+

an−1

a−1 b

which converges whenever 0< a <1, regardless ofx0:

lim

n→∞xn=

1

1−ab 0< a <1 ∀x0

Consequently the estimated fraction of active units corrected for multi-unit clusters is:

α0= ( 1 1− Ω LU )( N LU) = N LU−P sNsOs (4.9) Although I began by thinking of the multi-unit correction as an increase to the number of units I found, mathematically speaking it is a reduction in the number I expect to find: the term LU

in the denominator is the number of units I think I would record if all were active (the number of locations multiplied by the number in range at each location), and I am reducing it to account for those units I think I would not find because they would be clustered together. The requirement for convergence is

P sNsOs

LU <1

In practice the criteria is easily met. Note that usually Os ≤ 1, meaning that for realistic

voltage ranges there is less than 100% overlap with the same ranges of neighboring cells; U > 1, as will be shown in section 4.3.6, there are more than one spiny neuron in electrode range; and

L > P

sNs, because on average I record less than one unit per location, as described in section

4.3.1. In practice the convergence criteria was around 0.15-0.2 for my data in all the layers of cortex. Therefore I conclude that I can calculate the corrected fraction of active units using equation 4.9 without needing an initial “guess” for the active fraction and without actually applying the iterative procedure at all.