Norma Internacional de Información Financiera 1

The sensory layer is formed by a series of np neural populations NP1, NP2 … NPnp, each

of them with a preferred sensitivity to a certain stimulus magnitude.

Arbitrarily, we have established that the sensory layer for the variance experiment has np = 18 neural populations, each with a maximum sensitivity (Smax1, Smax2, Smax3, … Smaxnp) to 18 stimulus magnitudes evenly distributed along the (logarithmic) perceptual space: thus, preferred stimulus magnitudes are (in log space): 6.21 – 6.38 – 6.55 … 9.11.

Tuning functions

Each neural population has a tuning curve F characterizing its specific (yet probabilistic) sensitivity to certain stimuli. Following Jazayeri et al. (Jazayeri & Movshon, 2006), this

tuning function is defined per trial as the product of the baseline tuning function, Fo(NP)

Baseline tuning function

The baseline tuning function (of a neural population NP) is defined as a probability density function characterizing how likely is a neuron of that population to generate a signal in response to each stimulus of the perceptual space. Conversely, this tuning function also represents, in presence of a response from that neuron (a firing signal), the likelihood that such response has been produced by each of the possible stimuli of the perceptual space – by Bayes’ rule and disregarding constant terms (the PDF is normalized so that it sums to 1).

Specifically, the baseline tuning function of NP is a Gaussian PDF defined over the whole perceptual space (see ‘Stimuli’ section above), with mean at the stimulus with maximum sensitivity (Smaxnp) and standard deviation σtuning , wherein the latter is a free parameter

of the model representing sensory precision, which takes the same value for all neural

populations: (3) 𝐹_A(𝑁𝑃) = 𝑁6𝑆_KL0GM, 𝜎_NOGFGB: = 1 √2𝜋 ∗ 𝜎NOGFGB ∗ 𝑒/ 60/<QRSTU:1 2∗3_VWTXTY1

for x ∈ perceptual space.

Gain

As previously stated, the tuning function of each NP is the product of this baseline tuning function by a gain factor that changes on a trial-wise basis:

(4)

𝐹_G(𝑁𝑃) = 𝑔𝑎𝑖𝑛_G(𝑁𝑃) ∗ 𝐹_A (𝑁𝑃)

For the first trial of the experimental session, gain1(NP)=1 for all neural populations; this

value is established arbitrarily, and indicates that at the onset of the experiment we assume that the sensory layer is unbiased along the entire perceptual space.

The gain of each NP is subject to a decrease driven by the neural activity of the sensory layer, i.e., by the neural response of all NPs in response to stimuli. This gain decrease is the result of the addition of the decrease due to the activity of all (np) neural populations in response to all stimuli up to the previous trial, with each fractional effect scaled by perceptual and temporal distance.

Let us consider the gain factor of NP at trial n (gainn (NP)). It will be the sum of (n-1)*p

fractional effects, where n-1 is the number of trials prior to the current one and p is the number of neural populations of the sensory layer. In each trial, a stimulus Sn’ has been

received and each neural population has generated a population response R(NP’, Sn’),

which will have its effect on the current trial gain. Note that we are using Sn’ for the

stimulus received in trial n’ (n’< n) which elicits a certain response in the neural population NP’_{. In turn, this response will have an effect on the gain of the neural}

population NP at trial n and thus will modulate the response of NP to stimulus Sn, R(NP,

Sn). Later on we will see how neural responses are computed. We assume that the

response of the current trial does not have an instant effect on sensory gain; rather, the gain effect exerted by R(NP’, Sn’) on NP is maximal after stimulus Sn’ offset and decreases

afterwards. It is also maximal when NP=NP’ _{and decreases with perceptual distance.}

Unlike with trials, in which we assume the current response does not instantly change the gain in the current trial, all neural populations exert an effect on NP, including NP itself –but not instantly.

Before applying temporal discount, the maximal fractional effect on gainn(NP) exerted

(5) 𝑔_{^_`}_𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛_𝑚𝑎𝑥(𝑁𝑃, 𝑁𝑃h_{, 𝑛}h_{) = 𝑎𝑚𝑝} BLFG∗ 𝑆𝑑𝑢𝑟Gl ∗ 𝑅(𝑁𝑃h, 𝑆_Gl) ∗ m √2n∗3YRXT∗ 𝑒o(pQRSTUopQRSTUl)11∗qrstu1 where

1. ampgain is a free parameter which scales the maximal gain reduction (before

applying temporal discount, and with zero perceptual distance) and is constant for all NPs and trials.

2. Sdurn’ is the duration of the stimulus of trial n’. Thus, we assume that the sensory

response is produced in the sensory layer for the duration of the stimulus, or at least it is proportional to such duration, and the gain decrease is also proportional to the duration of the neural response that produces such effect. 3. R(NP’_{, Sn’) is the neural response of population NP’ when receiving stimulus Sn’.}

See below for details.

4. Smaxnp and Smaxnp’ are the preferred stimuli for which populations NP and NP’,

respectively, have maximum sensitivity; in other words, they are the stimuli at which the tuning function of the populations NP and NP’ peak.

5. σgain is the standard deviation of the gain field: a free parameter that modulates

the breadth of the effect of each neural response on the gain reduction of other populations.

In summary, the activity (neural response) elicited on a population NP’ by the stimulus Sn’ presented in trial n’ will produce a reduction on the gain of a neural population NP –

single neural population, NP’. This fractional gain-reduction is maximal right after offset of the stimulus Sn’ that elicited the activity, and experience time-discount afterward. The

maximal fractional gain-reduction effect is positively associated with the population response triggered on NP’ by Sn’ and on the duration of stimulus Sn’, and shows a

Gaussian decrease with perceptual distance between the preferred stimuli of NP’ and NP, scaled by the free parameters ampgain and σgain (amplitude and standard deviation

of the Gaussian gain function).

However, the stimulus presented in trial n’ elicits a neural response not only in one neural population, but in all p populations. All of them exert an effect on NP (including NP itself). We consider that the maximal gain effect produced by the entire sensory layer on NP, as a result of the activity elicited by Sn’ is the sum of all fractional effects:

(6)

𝑔^_`_𝑚𝑎𝑥(𝑁𝑃, 𝑛h) = v 𝑔𝑟𝑒𝑑_𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛_𝑚𝑎𝑥(𝑁𝑃, 𝑁𝑃hF, 𝑛h) M

Fwm

This effect is maximal right after Sn’ and presents a Gaussian-shaped decrease with time.

Thus, it will affect the population response to a later stimulus Sn (n>n’) in an amount

dependent of the elapsed time between trial n’ and n. Specifically, the gain-reduction effect of Sn’ on NP at the moment in which stimulus Sn is received is in function of the

maximal gain effect, the time between Sn’ offset and Sn onset and the free parameter

σrecovery, which modulates the decline of the gain effect with time. As the other free

parameters, σrecovery takes a single value for each participant and experiment. We will

explain later why this parameter is termed ‘recovery’. Formally: (7) 𝑔^_`(𝑁𝑃, 𝑛, 𝑛h) = 𝑔^_` _𝑚𝑎𝑥(𝑁𝑃, 𝑛h) ∗ 1 √2𝜋 ∗ 𝜎^_xAy_^z ∗ 𝑒 /(NFK_(Gh, G))1 2∗{|}~•€}|•1

However, the gain of NP (in trial n) is not only affected by a single trial, but by all trials up to n. We consider that the total gain-reduction effect on NP in trial n is the sum of all gain-reduction effects due to all trials up to n (all n’ ∈ 1 … n − 1 ), each one scaled by the temporal distance between that trial and n.

Formally: (8) 𝑔_{^_`</sub>_𝑎𝑙𝑙(𝑁𝑃, 𝑛) = v 𝑔<sub>^_`}(𝑁𝑃, 𝑛, 𝑛h₎ G/m Ghwm

where gred_all designs the gain-reduction effect exerted on a single neural population,

NP, by the neural activity of all neural populations due to all the stimuli received previous to trial n. This reduction is scaled by perceptual-space distance (to the preferred stimulus of NP) and by temporal distance (of the effect of each trial in experimental history).

Because we have established before that the initial gain of each neural population is Go=1, the gain of NP in trial n will be:

(9)

𝑔𝑎𝑖𝑛_G(𝑁𝑃) = 1 − 𝑔_{^_`}_𝑎𝑙𝑙(𝑁𝑃, 𝑛)

with a caveat: gain can take any value between 0 and 1 (both included), but cannot be negative. The model doesn’t admit negative (inhibitory) neural responses in the sensory layer.

(10)

𝑖𝑓 𝑔𝑎𝑖𝑛_G(𝑁𝑃) < 0 → 𝑔𝑎𝑖𝑛_G(𝑁𝑃) = 0

Because the effect of each trial in experiment history (n’ ∈ 1 … n − 1) declines with time, when examining the effect that one individual trial has on subsequent iterations,

we observe a recovery of the gain-reduction caused by that specific trial. This is why σrecovery is considered to measure the rate of gain recovery.

Tuning function

As a recap, the tuning function of a neural population NP in trial n is the product of its baseline tuning function by the gain factor of NP (which changes on a trial-wise basis, as we have seen):

(4)

𝐹G(𝑁𝑃) = 𝑔𝑎𝑖𝑛G(𝑁𝑃) ∗ 𝐹A(𝑁𝑃)

where 𝐹_A(𝑁𝑃) = 𝑁(𝑆_KL0GM, 𝜎_NOGFGB), a Gaussian PDF defined over the perceptual space.

Thus, the value of the tuning function of NP at one specific stimulus magnitude presented on trial n (Sn) will depend on three factors:

1. The distance between the maximum sensitivity stimulus of NP (Smaxnp) and the

presented stimulus Sn.

2. The value of σtuning,which is a free parameter for adjusting sensory precision, i.e.

internal sensory noise. This value is assumed to be equal for all neural populations.

Neural response

Neural response of a single neural population (population response)

The neural response elicited on a neural population NP by a stimulus Sn presented in

trial n, termed R(NP, Sn), is given by the value of the tuning function of NP at Sn:

(11)

𝑅(𝑁𝑃, 𝑆G) = 𝐹G(𝑁𝑃, 𝑆G)

As stated before, the value of the tuning function depends of the distance between Smaxnp and Sn, σtuning and gainn(NP).

This is why σtuning is a measure of sensory precision, or internal noise: if σtuning is very

small, the value of the tuning function of any neural population whose maximum sensitivity is not very close to Sn will be very low, and only the neural population that is

highly tuned to that stimulus will contribute significantly to the overall response of the sensory layer: thus, the overall sensory response will be very ‘crisp’ and neatly tuned to the veridical stimulus. Conversely, if σtuning is large, the value of the tuning function of

most neural populations will be large enough to contribute significantly to the overall response, even if their peak sensitivity is far from the current stimulus level. In this case, the overall sensory signal will be less precise – less informative regarding the stimulus that originated it.

For simplicity, our model does not consider external noise: each RDK provides one unambiguous variance stimulus, given by the dispersion of the direction of its components –even if variance itself is a statistical property related to the precision of an ensemble.

Overall sensory response – likelihood distribution

Following Jazayeri et al. (Jazayeri & Movshon, 2006), the logarithm of the probability distribution representing the neural response elicited by the entire sensory layer (when receiving Sn) is given by the weighted average of the logarithms of the tuning functions

of all neural populations, each one weighted by its neural response. Formally: (12) log6𝑅(𝑆_G): = v 𝑅(𝑁𝑃_F, 𝑆_G) ∗ log6𝐹_G(𝑁𝑃_F): M Fwm

This probability distribution is a function defined along each considered value of the perceptual space. We now normalize it by the sum of all its values along the perceptual space so that it sums to one:

Let ps be the number of considered values in the perceptual space (90 in our experiment), xi each of those values and SumR(Sn) the sum of all values of R(Sn) for all xi

in the perceptual space:

(13) SumR(Sn) = v 𝑅(𝑆_G)(𝑥_F M• Fwm )

However, previously we have calculated log(R(Sn)) instead of R(Sn) directly.

(14) SumR(𝑆_G) = v 𝑅(𝑆_G)(𝑥_F M• Fwm ) = v 𝑒Ž•• ("(<T)(0F)) M• Fwm Therefore:

(15)

log (𝑅_GA^K(𝑆_G)) = log • "(<T)

<OK"(<T)‘ = log(𝑅(𝑆G)) − log6𝑆𝑢𝑚𝑅(𝑆G): = log (𝑅(𝑆G)) −

log (∑M• 𝑒Ž•• ("(<T)(0F)))

Fwm

The normalized response of the entire sensory layer after reception of a stimulus Sn is

what we call the likelihood distribution: Lkn = Rnorm(Sn), representing the probabilistic

sensory response to stimulus Sn, defined at each value of the perceptual space.

(16)

𝐿𝑘G = 𝑅GA^K(𝑆G) = 𝑒Ž•• ("T•–Q(<T) )

The mean and standard deviation of the likelihood distribution are calculated weighting values by their respective probability:

(17) 𝜇_˜™G = v 𝐿𝑘_G(𝑥_F) M• Fwm ∗ 𝑥_F (18) 𝜎_˜™G = šv 𝐿𝑘_G(𝑥_F) ∗ (𝑥_F − 𝜇_˜™G)2 M• Fwm

In normal conditions (i.e. unless gain-related parameters take highly deviant values), the likelihood distribution can be approximated to a Gaussian shape:

(19)

𝐿𝑘_G ≈ 𝑁(𝜇_˜™G, 𝜎_˜™G)

For simplicity, in the next steps we will treat the likelihood as a Gaussian probability density function computed over the perceptual space.

This likelihood distribution is the neural response generated in the sensory layer by the stimulus Sn: in other words, the output of the sensory layer as produced by the model,

which is then transferred forward to the decision layer.

In document Diseño de un sistema administrativo contable en la empresa comercial "Niño Edwin" ubicado en la ciudad de Santo Domingo d elos Tsáchilas período 2013 (página 49-54)