The dynamics of cortical hypercolumn in visual area was described by ring model (Ben-Yishai et al., 1995; Hansel and Sompolinsky, 1998), which is rate model (Wilson and Cowan, 1972; Equation 1) with special connectivity matrix described by the (Equation 2) below. We also introduce short term depression as in (Tsodyks et al., 1998; Romani and Tsodyks, 2014; Equation 3).

Here m_i(t) is a mean firing rate of the unit i, τ is time constant, g is a gain function, I₀ is a mean depolarization level. Each i-th unit has its preferred angle θi=iπN which corresponds to preferred orientation (PO). W_ij is the weight between i-th and j-th units, given by Equation (2), it includes homogeneous inhibitory part J₀ and spatially modulated part J₁cos(2(θ_i−θ_j)) which represents stronger connectivity between units sharing similar POs. N is the number of units.

U is a baseline level of neurotransmitter release probability. τ_rec is a time constant for neurotransmitter recovery. x_i(t) is an amount of neurotransmitter in presynaptic unit i, governed by the following equation:

Iiinp(t) is the external stimuli input for unit i, with amplitude C and duration T. Angles θkinp of external stimuli are uniformly distributed between 0 and π. rect is rectangular function, (Equation 5). N_stim, t_k stand for the number and times of stimuli, respectively:

Finally, Iinoise(t) is a colored noise with standard deviation σ and correlation time τ_n, each unit receives noise uncorrelated with others:

Parameters for simulations are shown in the Table 1. These parameters are used in all simulations unless otherwise stated.

Transitions between functional states were implemented by simultaneously changing U and I₀ while keeping mean firing rate ⟨m⟩ ≈ 0.5 Hz, (see Equation 7). T_sim is total simulation time. Corresponding values of I₀ for each U are shown in Figure 2.

In order to analyse the detection precision of orientations of external stimuli we calculate the so called Population Vector (PV), (Equation 8). The PV is a complex number which represents the angle of activity bump (argument of PV) and its magnitude (absolute value of PV). Readout was performed with variable numbers of neurons N_read. Readout uses a sample of units than are involved in processing of stimuli, reflecting sparse connectivity between different regions of the cortex (Schüz, 2003).

where R(t) population vector, χ_j(t) is a number of spikes emitted by neuron j, chosen randomly with Poisson distribution those probability mass function is P(k)=(mj(t))kk!e-mj(t).

To distinguish the errors caused by sparseness and discreteness of readout and the errors caused by moving of bumps we also calculate PV, according to Equation (9). ER (Exact Readout) can be considered as 2nd coefficient of spatial Fourier expansion of activity m_j(t).

For each stimulus deviation of detected angle (argument of R(t) and ER(t)) from the true angle of the stimulus was calculated and averaged to estimate detection error (Equation 10). In error estimation we also introduce a delay (lag), because it takes time for readout neurons to react. Hence, for each functional state controller by U, I₀ and readout sparseness N_read we find lag for which detection error is minimal.

Dynamical system has been integrated by Euler-Maruyama method for SDE with 0.002 s time step. Calculation script was written in Python RRID:SCR_008394 with the help of numpy RRID:SCR_008633 and matplotlib RRID:SCR_008624; (Hunter, 2007) libraries. To run calculation for different parameters in parallel GNU parallel utility has been used (Tange, 2011). Scripts can be downloaded from github repository https://github.com/esirpavel/ring_plasticity_V1_fcn.

In recurrent networks with ring architecture, spatially tuned activity can appear if self-excitation is strong enough (Ben-Yishai et al., 1995; Hansel and Sompolinsky, 1998). Further we will call such kind of activity “bumps.” When effects of synaptic plasticity are taken into account, repertoire of possible activities becomes richer and can lead to generation of waves, or spontaneous bumps of activity propagating short distances and then disappearing (York and van Rossum, 2009; Romani and Tsodyks, 2014).

Experimental evidence indicates that mean firing rate remains approximately constant for different behavioral states if no sensory stimuli is presented (Poulet and Petersen, 2008). To achieve this for each value of the release probability U, we set such value of baseline depolarization I₀ that mean firing rate (Equation 7) remains approximately equal to 0.5 Hz in the absence of external stimuli (C = 0). Varying I₀ and U simultaneously emulates influence of acetylcholine (ACh) or other neuromodulators that can reduce release probability and at the same time enhance depolarization of membrane potential by changing conductance of potassium channels (McCormick et al., 1993; Giocomo and Hasselmo, 2007).

Activity for several characteristic values of U and I₀ are shown in Figure 1. When increasing baseline release probability U, spontaneous bumps appear at U ≈ 0.4, further increasing their amplitude with growing U. All values of I₀ and U that were used in simulations are shown in Figure 2. The positive slope in the beginning of the curve in this figure may appear paradoxical but can be understood as follows. When U is small, noise does not cause significant spatial fluctuations in the network activity, and the inhomogeneous input to the i-th neuron r1i=J1U∑jcos(θi - θj)xj(t)mj(t) is smaller than homogeneous inhibition r0=J0U∑jxj(t)mj(t), because convolution with cosine filters any spatially unmodulated input. Therefore, increasing U leads to decreasing of synaptic input to the neurons and hence for keeping the network activity at the same value we should increase I₀. But, increasing U after reaching some intermediate values (in this case it is ≈ 0.2) noise starts to induce spatial fluctuations comparable with mean activity and hence excitation r1i dominates over inhibition r₀ because |J₁| > |J₀| (note than J₀ is negative while J₁ is positive). Thus, further increasing of U requires decreasing of the baseline input I₀ to maintain the desired average network activity.

Each presented stimulus is characterized by time of occurrence and orientation angle. The input to each unit is set according to Equation (4), so units with preferred angles matching the orientation of the stimuli receive the strongest input with amplitude C. Duration of stimuli is T. Stimuli were presented as Poisson process with frequency Freq = 4 Hz and refractory period T in order to eliminate intersections. Thus, we do not model continuous presentation of stimuli, but only brief ones.

Figure 3 illustrates the network activity in the presence of external stimuli. Each stimulus triggers a bump and changes the angle of PV toward the angle of the external stimulus (red horizontal lines on the left and right side). However, detected angles and applied ones do not match entirely. This happens because of two reasons. The first one is variability of neuronal response caused by discreteness and sparseness of readout; and the second one is intrinsic dynamics of the network. For small values of release probability firing rate is small and error caused by variability of neuronal response is large. It can be seen from Figure 3A, where sparse readout angle fluctuations around the stimulus are large, and at the same time exact population vector angle is very close to stimulus angle. For large values of U (Figure 3C) dynamics of the network becomes unstable and after presenting the stimuli a moving bump of activity can emerge, and it takes time before the angle detected by the network will settle down to the presented one. Between two of this regimes is an optimal state at which an error of sparse readout is minimal (Figure 3B).

We now address the main issue of the paper, namely the precision with which the network detects the orientations of external stimuli for different functional states. To this end, we calculated the error of detection according to Equation (10), for different values of the release probability U and numbers of readout neurons N_read. Also we considered errors when PV was calculated according to Equation (9) (exact readout). The plots of the detection error vs. U, for small and large amplitudes of stimuli, are shown on Figure 4. On each graph, optimal detection for sparse readout is reached for some intermediate value of the release probability. Lags for minimal detection error for C = 20 are show on Figure 5. For bigger firing rates lags for optimal detection are smaller. Further all simulations was performed with T = 0.05 s and freq = 4 Hz. Simulations with longer and more frequent stimuli give qualitatively same results (data no shown).

The optimal value of U becomes lower while number of readout neurons is increased, similarly to what was reported in Mark and Tsodyks (2012). Also there is a tendency for the optimal U to be smaller for stimuli with higher amplitudes, as one can see in Figure 6. We discuss the implications of these observations below in Discussion section.

Appearance of optimal regime for orientation detection can be understood as follows. For large values of synaptic release probability, noise induces bigger fluctuations in network activity that in turn lead to moving bumps rotating away from the angle of the presented stimulus. This tendency is monotonic, as shown on Figure 4, dashed lines for exact readout. So firstly we have the tendency that smaller values of U are better for detection. But on the other hand, network with small values of release probability generates activity with lesser firing rate peaks and thus activity of readout neuron R(t) is more sensitive to noise, especially when N_read is relatively small. Competition between these two tendencies lead to the emergence of the optimal value of U for detection. According with the last tendency, the sparser readout is, the more dominating the effect of noise on readout, and hence the greater the optimal value of U for minimal error should be, consistently with our findings (Figure 6). Similar reasoning explains why increasing the amplitude of the stimuli should also result in smaller optimal values for release probability U, for bigger amplitudes, e.g., C = 40 and full readout, dependence of detection error from U becomes almost monotonically increasing as for exact readout (data not shown).

In Pinto et al. (2013) it was shown that optogeneric activation of basal cholinergic neurons or their axons in V1 improves visual discrimination in awake mice. Comparing with our finding, we can hypothesize that in normal conditions in quiet animal when sensory stimuli are sufficiently strong and baseline release probability is relatively big, for better performance when animal beginss to engage with environment, baseline release probability should be reduced by releasing of (ACh).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.