Mean frequency

Neurons communicate mainly by transmitting spike trains (an exception is the photorecep- tor which transmits photic information to ganglion cells by mean of graded potentials). The in- terspike interval is known to be linearly related to stimulus current intensity1 with two devia- tions (Connor and Steven 197 1): (i) for strong currents, a saturation occurs because of the re- fractory period of the cell which limits spike frequency; (ii) for weak currents, the cell threshold may not be reached and thus no spikes are emitted. These characteristics - saturation, linear range and threshold - can advantageously be described by one of the following sigmoid-like functions:

1 1

f(x) = - . ( 1

+

^tanh ( g x ) ) = ----

2 I

+

^e-2gx

In these equations, f(x) stands for an average rate of firing or mean frequency, generally normalized to 1. Variable x represents the mean soma2 potential (and not the current). Corre- spondence between current and potential may be established by Ohm's law ($3.1.3). Equation (4.1) is so-called "Hill function". For n > 1, this function has a sigmoid shape, and for increas- ing values of n, sigmoids become steeper with the inflection point closer to x = 8. The limit case, n

-+

=, is well represented by a step function (see equation (3.40)).

A particular case of equation (4.1) is given by equation (4.2) where b determines the point where the slope is maximum3. Equations (4.3) and (4.4) are directly related. g is the gain rep- resenting the slope at the inflection point and xo is the spontaneous firing of the cell. Sponta- neous firing expresses the property of neurons to emit spikes even without stimulations. Graphs of equations (4.2) and (4.3) are shown in Figure 4.1.

1. The relationship frequency-intensity varies with time for constant stimulus currentTypically, there is a lengthening of interspike intervals as the splke train proceeds. This phenomenon reflects the presence of spike frequency adaptation (Connor and Steven 1971; Yamada et dl. 1989; Getting 1989).

2. The soma is the cell body,

3. This case occurs for T f l x x ) _ax = 0, and thus, x =

&

By using an average rate of firing, determined on the basis of the mean soma potential, it is sought to avoid a microscopic analysis where contributions of every spike on every synapse would have to be calculated. Such a macroscopic analysis may also be understood from a sta- tistical point of view. Instead of considering output values as mean firing rates, it may be inter- preted as the probability of the emission of a spike during a fixed time interval.

Figure 4.1 : Continuous relationship between mean soma potential CMSP) and hquency. (A): Equation (4.2) with a = 1, b = 3 . (B): Equation (4.3) with xo = 0.5 and for several values of gain g : 1 .O, 2.0, and 4.0

4.1.3 Cell latency: the leaky integrator

In CHAPTER 3 a neuron was shown to be composed of a capacitance and of a set of con- ductances whose one, a passive conductance, represented leakages in the membrane cell. Such a neuron is now referred to as a leaky-integrator (leaky for the passive conductance, integrator for the capacitance effect). By considering neuronal information as being coded by spikes reg- ularly spaced, and neurons to be leaky-integrator elements, this paragraph will show how these two assumptions can readily explain the dependence of the cell reaction time upon stimulus amplitude. Furthermore, this dependence will be compared to the relationship existing between luminance intensity and photoreceptor reaction time.

Mean frequency, presented in

5

4.1.2 and interpretable in term of probability, has one ma- jor drawback, raised by Hopfield (1984): "This stochastic view preserves the basic idea of the input signal being transformed into a firing rate but does not allow precise timing of individual action potentials." To retrieve this information, a microscopic analysis must be achieved. In particular, temporal properties and the threshold of the cell need to be known. Formalization of this problem is made for the leaky-integrator, followed by a comparison with a real case, ob- tained from an experimental study on photoreceptors.

Intensities of quantitative stimuli are usually transmitted by periodic or nearly periodic spike trains (with exceptions found in the retina), represented schematically in Figure 4.2.

Thus, the period can be formalized as the spike duration plus the interspike duration. A second formalization is to consider neurons as being subdivided into two functional stages, as shown in Figure 4.3.

U H M ~ kK 4. l'emporal Analysis

"ff

Figure 4.2 : Intensity translated into spike frequency. ton is the duration of the spike and is supposed to be con- stant The frequency f = (ton

+

tuff)-' is a monotonic function of the intensity: f = a . A

+ P

and yields tof! = ((a - A + 6 )

-'

^{- ton with} a a positive constanant Note that even a zero intensity is converted into a spike tram of period

P

+ to,.

Effect of the capacitance and of the passive conductance are represented, respectively, by an integration during to, and a leakage during tofy The equivalent electrical circuit is shown in Figure 4.4. It is composed of one passive conductance gleak, one time-dependent conduc- tance g,,,, a membrane capacity C and two voltage sources Eleak and E,,,. The time-depen- dent conductance, activated by spikes impinging on the cell excitatory input, is serially linked to the positive voltage source E,,,. The current generated by this branch is injected into: (i) the capacitance and (ii) the leakage pathway. In this latter pathway, the passive conductance gkQk is serially linked with the negative voltage source Eleak.

integration energy decay

Figure 4.3 : Functional stages of a leaky integrator cell. There are two modules corresponding to two stages: (1) integration during ton; (2) energy decay during tuff (see Figure 4.2 for definition of ton and tuff).

Figure 4.4 : Electrical circuit of the cell model. For the theoretical analysis, the excitatory conductance is ruled by the all-or-nothing principle: during ton, ge,,= 0.28 pS; during tofP g,,,= 0. For simulation, excitatory conduc- tance varies according to the alpha function (equation (3.2)). In both cases, excitatory potential EeXc= 20.0 mV, Eleak= -71.0 mV, gleak= 0.08 pS, C= 2 nF, threshold = -45.0 mV. The electrical parameters are inspired from those found in Wehmeier et al. (1989). Conductances are measured in siemens, noted S.

Neuronal latency: mathemaQca1 evidence o 1

According to Figure 4.2, spikes are supposed to last a fixed amount of time ton , the interval between spikes

r

being dependent on the mean frequency1. In order to ease the temporal

o f f

analysis, a simplifying assumption is made: when receiving a spike, the time-dependent exci- tatory conductance has a constant positive value (during ton) and a zero value in between spikes (during toff). A more realistic model would require a time varying conductance based on the alpha synaptic function (equation (3.2)). Simulations indicate, however, that both mod- els yield identical qualitative results (compare curves A to curve B in Figure 4.5).

When a spike train impinges on the cell excitatory input, the membrane capacity integrates pulses through conductance g,,, during ton, while the stored energy leaks through conduc- tance gleak during top A temporal analysis of this circuit shows that succeeding pulses at the entry cause the output potential to oscillate while rising steadily towards an asymptotic value.

This value is a monotonic function of the spike frequency (see Appendix 7 for equations). A more pertinent problem for this thesis is to determine the precise instant at which the potential reaches a fixed value, representing a static cell threshold. The time difference between this in- stant and the appearance on the postsynaptic membrane of the stimulation is the latency of the cell. The mathematical development for determining the relationship between spike frequency and latency is to be found in Appendix 7.

The number of period p of a spike train of frequency f required for a threshold s to be reached by the membrane potential is given by:

p = - ²² . log

'van

⁺ 71tOff

The time constants ^{T ~ , Z~} and the constants E , Eleak, k, k' are defined in Appendix 7. Inter- spike interval tog is related to spike frequency f by:

toff =

7

1 ^{- t o n}

A period lasting ton

+

top cell latency becomes:

Figure 4.5 shows cell latencies computed for spike trains with frequencies ranging from 1 to 800 Hz. From this figure, it can be observed that higher spike frequencies yield shorter laten- cies. Latencies increase rapidly for low frequencies and under a certain critical frequency the threshold will never be reached. Inversely, for the upper half range of frequencies latencies are short and decrease slowly. Cells with short latencies will be said to have the temporalprece- dence over cells with longer latencies (see definitions in

3

4.4.2). This notion of precedence has also been used in Kurogi (1987) where the so-called function "maximum detection" detects

1. With stochastic models for spike trains, there is a standard deviation on the interspike interval (Sam- path and Srinivasan 1977; for the leaky integrator cell, Bugmann 1992).

0 o CHAP 1 bK 4. 1 emporal Analysis cells whose potentials reach the threshold value first.

loo] A

200 400 600

800

Frequency (Hz)

Figure 4.5: Latency of the electrical model for periodic spike trains (electrical parameters as in Figure 4.4). The curve labelled (A) has a time-varymg conductance based on the alpha synaptic function; h s curve was obtained by simulation (Wilson and Bower, 1989; parameters are: z= 1.0 ms; G = 0.28~s). The curve labelled (B) has the simplified time-varying conductance presented in the text, and was BGknnined using equations (4.5) and (4.7).

For curve (B), below 89H2, the potential does not reach the threshold and so latencies are undefined. The critical frequency (see text) is therefore 89 Hz with a latency of 97 ms.

4.1.4 Cell latency: the photoreceptors

The vertebrate retina has two kinds of photoreceptors with differing spectral properties: the cones and the rods. Cones and rods are specific neurons sensitive to light. They perform the phototransduction by converting photons into electrical signals. Knowledge of the biochemis- try of the phototransduction results mainly from research on rods but cones underlie the same kind of mechanisms (see recently Schnapf et al. 1990).

Rods are composed of two p r t s l : the inner segment whose structure is similar to any other neuron and the outer segment which contains all the machinery needed for the detection of pho- tons. The outer segment is a tube within which some 2000 disks are stacked. Between the mem- brane of both tube and disks there is no continuit? and communication necessitates a chemical messenger, the cGMP. Despite their similar structures the tube membrane and the disk mem- brane have distinct functions: the first contains the rhodopsin3, a molecule photosensitive, while the latter contains the different ionic channels necessary to produce electrical signals.

Phototransduction is a complex process divided into three main stages: (i) activation of the rhodopsin to a photon4; (ii) the active rhodopsin triggers a cascade of chemical reactions result-

1. This description is based on a description from Bader (1988).

2. The cone outer segment consists of a single membrane folded back and forth and thus there is no such discontinuity (Knowles 1982).

3. A human outer segment has about one billion rhodopsin molecules distributed in the disks.

4. The rhodopsin is constituted of one protein, the opsin, and one organic substance derived from vitamin A, the 11 -cis-retinal. The absorption of a photon by the molecule 11 -cis-retinal leads to an isomerization which then affects the opsin molecule. Isomerization corresponds to a change in rhe form of molecular structures.

Neuronal latency: mathematical evidence by

ing in a concentration diminution of the chemical messenger cGMP; (iii) reduction in the cGMP concentration diminishes the amount of opened ~ a ' channels with the issue that the cell becomes hyperpolarized.

Latency, not surprisingly, depends upon the various chemical reactions involved during these three stages. Their mathematical formulation being not obvious, only experimental data are now considered to introduce latencies. In this kind of experiment, stimulus amplitude is de- termined by the intensity of a light source flashed briefly. Retinal activity, resulting from flash stimulations, is then measured through electroretinograms. Delays between flash activations and retinal activity variations include various neuronal interactions in addition to the photore- ceptor stage. Nevertheless, Mansfield and Daugman (1978), using such recording techniques, have concluded that, in the retina, the dominant latency component lies in the photoreceptors.

To describe the latency measurements as of function of flash intensity, they proposed the fol- lowing equation:

This equation can be rewritten as:

where I is the intensity of a short flash (10 ms), E the characteristic latency of signal response, lmi, the minimum latency (asymptotic value),

P

⁼ 1 /3, and k is a constant. Qualitatively, this relationship latency-intensity, illustrated in Figure 4.6, is comparable to the one obtained from the leaky-integrator model.

0 ;

^I

^I

^I

ⁱ

0 250 500 750

loo0

Intensity

Fisure 4.6 : Illustration of the relationshtp latency-intensity expresses by equation (4.9). Parameters are:

lmi, = 10 ms, k = 0.2, P = 1/3.

I u LHllr 1 k K 4. 'l'emporal Analysis

4.1.5 Latency in the ~imulus' eye: the Fuortes-Hodgkin model

The Fuortes-Hodgkin model (described in Fuortes and Hodgkin 1964) is an electrical cir- cuit composed of n serial but independent stages2 (Figure 4.7) whose role is to mimic the dif- ferent stages in the ommatidia3 of Limulus, from the absorption of photons to the generation of nerve impulses. In this electrical model, there are units, labelled p, which amplify and iso- late one stage from the others. Thus, capacitances C and resistances R constitutes separate leaky-integrator circuits (RC electrical circuits). Leakage conductances g = 1/R are sup- posed to increase linearly with the output vn :

where go and h are constants.

Figure 4.7 : Fuortes-Hodgkin model of the Limulus eye. This electrical circuit is composed of n stages isolated from each other by amplifiers p. The conductance g varies linearly with the output potential vn. C is the capac- itance, and vo the input potential, proportional to luminance.

This electrical circuit is now analyzed. For the r' stage, the equation describing current variations is:

which can be rewritten as

with T~ = C/go. At equilibrium4 dv

-'

⁼ 0. Using this equilibrium state for every value of r, d t

1. Limulus or horseshoe crab.

2. The number of stage has been estimated by various authors. Accorang to Donner (1989), n = 6 - 8 is suggested not only by frog ganglion cells and human psychophysics, but also by studies on the macaque. To be in accordance with the Pulfrich effect, Alpern (1968) found n = 4. For the Limulus, Fuortes and Hodglun (1964) used values of n greater than 6.

3. An ommatidia is one of the elements corresponding ^t a small simple eye that make up the compound eye of an arthropod (definition from the Webster dictionary).

4. Development inspired from Alpern (1968).

Neuronal latency: mathematical evidence 7 1 comprised between 1 and n, the input potential vo can be expressed as a function of the output potential v,:

vo = v, ( 1

+

^v,/h)

"

^{(go/p) (4.13)}

For a luminance I ten times above the absolute cone threshold, it is possible to make the as- sumption that ( 1

+

v,/h) does not differ significantly from (v,/h) (Alpem 1968). Also, since vo is supposed to be linearly related to luminance (v, = I/b), equation (4.13) becomes:

and thus v, is given by:

In this circuit, the time constant is t = R C = C/g . With the above luminance assumption, g can be approximated by g = go (v,/h) . Also, to = C/go and thus

2 = zo (X/v,) (4.16)

Also, response v, to a step of luminance I is given by (Fuortes and Hodgkin 1964):

As in

5

4.1.3, it is wished to establish the relationship between luminance intensity and the latency required for a specific potential, representing a threshold, to be reached. Thus, time t in equation (4.17) must be extracted. This operation being non-trivial, equation (4.17) is linear- ized, resulting in relation (details can be found in Appendix 8):

where v, stands for the cell threshold, and m, xo are defined in Appendix 8. Substituting t in this equation by its definition (4.16) and using equation (4.15) for v, , latency 1, replacing time

t in equation (4.18), becomes:

where K' is defined in Appendix 8.

The same development can be applied for a flash, instead of a step of luminance. The re- sponse v, to a flash of duration A t and of luminance I is given by (Fuortes and Hodgkin 1964):

By analogy with the method used for the response to a step, it can be shown that the latency,

/ 2 CHAPTER 4. Temporal Analysis with respect to a threshold v,, is given by:

where cl and c2 are constants. Note that Alpern (1968) has determined a particular case of this equation. Instead of calculating latency with respect to a threshold, he calculated the latency required for potential v, to reach its maximum. He obtained:

where K = ( n - 1) K'. It is worth noting the good accordance between equations (4.22) and (4.9). Furthermore, it can be observed that latencies defined with respect to a threshold for, re- spectively, a step (equation (4.19)) and a flash (equation (4.21)), are very similar. Furthermore, qualitatively, these equations are in accordance with the relationship luminance-latency seen previously for both the leaky-integrator model and the photoreceptors. This similitude can be seen when comparing the graph of equation (4.19), found in Figure 4.8, with the graphs shown in Figure 4.5 and Figure 4.6.

l-4

Intensity

0

0 250 500 750 1000

Figure 4.8 : Determination of the latencies of the Fuortes-Hodgkin model solicited with a step of light of varying luminance (equation (4.19)). Luminance range 0.. . 1 0 ; parameters are: n = 5, v, = 5 mV , A = l.6xlo7 (from Fuortes and Hodgkin 1964), K = 78 ms troland2 (from Alpern 1968). C w e (A) is determined using a nu- merical solution for t in equation (4.17). Curve (B) is determined using an approximation of the solution for t (with equation (4.18)).

4.2 Neuronal latency: techniques of measure

Dans le document Understanding the early human visual system through modeling and temporal analysis of neuronal structures (Page 89-97)