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Divisive gain modulation with dynamic stimuli in integrate-and-fire neurons.

Ly C, Doiron B - PLoS Comput. Biol. (2009)

Bottom Line: It has been shown that divisive gain modulation of neural responses can result from a stochastic shunting from balanced (mixed excitation and inhibition) background activity.However, input statistics, such as the firing rates of pre-synaptic neurons, are often dynamic, varying on timescales comparable to typical membrane time constants.Using a population density approach for integrate-and-fire neurons with dynamic and temporally rich inputs, we find that the same fluctuation-induced divisive gain modulation is operative for dynamic inputs driving nonequilibrium responses.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania, USA. chengly@math.pitt.edu

ABSTRACT
The modulation of the sensitivity, or gain, of neural responses to input is an important component of neural computation. It has been shown that divisive gain modulation of neural responses can result from a stochastic shunting from balanced (mixed excitation and inhibition) background activity. This gain control scheme was developed and explored with static inputs, where the membrane and spike train statistics were stationary in time. However, input statistics, such as the firing rates of pre-synaptic neurons, are often dynamic, varying on timescales comparable to typical membrane time constants. Using a population density approach for integrate-and-fire neurons with dynamic and temporally rich inputs, we find that the same fluctuation-induced divisive gain modulation is operative for dynamic inputs driving nonequilibrium responses. Moreover, the degree of divisive scaling of the dynamic response is quantitatively the same as the steady-state responses--thus, gain modulation via balanced conductance fluctuations generalizes in a straight-forward way to a dynamic setting.

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Parameter space where divisive gain modulation extends to the nonequilibrium regime.(A) The logarithm of the area (or error) between the time-dependent response curve scaled by the equilibrium scale factor  and the  curve for many background levels and many driver inputs: . The  values on the vertical axis corresponds to a scaling of the driver input used in Figure 3A (see text for details). The two points at  and , 1900 s−1 marked by stars (*) correspond to the difference in area between the curves in Figure 3D, and the two black circles (•) at , 1900 s−1 correspond to the difference in area between the curves in Figure 5B. Any region with colors in the range of orange to blue correspond to parameters where divisive gain modulation persists. (B) The average (unscaled) time-dependent response of the neurons with the same parameters and driver inputs as (A) are plotted on a logarithmic scale. The three stars (*) at  are the average firing rates of the unscaled responses in Figure 3C, and the three black circles (•) at  are the average firing rates of the unscaled responses in Figure 5A (bottom panel). A logarthmic scale was used to better highlight the variety of values.
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pcbi-1000365-g006: Parameter space where divisive gain modulation extends to the nonequilibrium regime.(A) The logarithm of the area (or error) between the time-dependent response curve scaled by the equilibrium scale factor and the curve for many background levels and many driver inputs: . The values on the vertical axis corresponds to a scaling of the driver input used in Figure 3A (see text for details). The two points at and , 1900 s−1 marked by stars (*) correspond to the difference in area between the curves in Figure 3D, and the two black circles (•) at , 1900 s−1 correspond to the difference in area between the curves in Figure 5B. Any region with colors in the range of orange to blue correspond to parameters where divisive gain modulation persists. (B) The average (unscaled) time-dependent response of the neurons with the same parameters and driver inputs as (A) are plotted on a logarithmic scale. The three stars (*) at are the average firing rates of the unscaled responses in Figure 3C, and the three black circles (•) at are the average firing rates of the unscaled responses in Figure 5A (bottom panel). A logarthmic scale was used to better highlight the variety of values.

Mentions: In our model, when the driving input rate is low, the population of neurons rarely fire action potentials (i.e., low spontaneous activity). The firing rates in our simulations in this state range from nearly 0 to 3 s−1, depending on the background level of activity. Although extracellular recordings in the cortex suggest the neurons can fire spontaneously at rates larger than 2 s−1 [43], such experiments are usually biased towards active neurons. Extracellular recordings by [44] that were unbiased towards responsive neurons suggest that many neurons have low spontaneous firing rates and that only a small fraction of neurons respond ‘well’ to stimuli in unanesthetized animals; this fact was also discussed in [43]. Moreover, calcium imaging experiments of awake and anesthetized rats in layer 2/3 of the cortex show that many neurons have resting firing rates less than 1 s−1 [45]. The actual firing rate of neurons in the resting state is a contentious issue, but our results hold for many parameter regimes (see Fig. 6).


Divisive gain modulation with dynamic stimuli in integrate-and-fire neurons.

Ly C, Doiron B - PLoS Comput. Biol. (2009)

Parameter space where divisive gain modulation extends to the nonequilibrium regime.(A) The logarithm of the area (or error) between the time-dependent response curve scaled by the equilibrium scale factor  and the  curve for many background levels and many driver inputs: . The  values on the vertical axis corresponds to a scaling of the driver input used in Figure 3A (see text for details). The two points at  and , 1900 s−1 marked by stars (*) correspond to the difference in area between the curves in Figure 3D, and the two black circles (•) at , 1900 s−1 correspond to the difference in area between the curves in Figure 5B. Any region with colors in the range of orange to blue correspond to parameters where divisive gain modulation persists. (B) The average (unscaled) time-dependent response of the neurons with the same parameters and driver inputs as (A) are plotted on a logarithmic scale. The three stars (*) at  are the average firing rates of the unscaled responses in Figure 3C, and the three black circles (•) at  are the average firing rates of the unscaled responses in Figure 5A (bottom panel). A logarthmic scale was used to better highlight the variety of values.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2667215&req=5

pcbi-1000365-g006: Parameter space where divisive gain modulation extends to the nonequilibrium regime.(A) The logarithm of the area (or error) between the time-dependent response curve scaled by the equilibrium scale factor and the curve for many background levels and many driver inputs: . The values on the vertical axis corresponds to a scaling of the driver input used in Figure 3A (see text for details). The two points at and , 1900 s−1 marked by stars (*) correspond to the difference in area between the curves in Figure 3D, and the two black circles (•) at , 1900 s−1 correspond to the difference in area between the curves in Figure 5B. Any region with colors in the range of orange to blue correspond to parameters where divisive gain modulation persists. (B) The average (unscaled) time-dependent response of the neurons with the same parameters and driver inputs as (A) are plotted on a logarithmic scale. The three stars (*) at are the average firing rates of the unscaled responses in Figure 3C, and the three black circles (•) at are the average firing rates of the unscaled responses in Figure 5A (bottom panel). A logarthmic scale was used to better highlight the variety of values.
Mentions: In our model, when the driving input rate is low, the population of neurons rarely fire action potentials (i.e., low spontaneous activity). The firing rates in our simulations in this state range from nearly 0 to 3 s−1, depending on the background level of activity. Although extracellular recordings in the cortex suggest the neurons can fire spontaneously at rates larger than 2 s−1 [43], such experiments are usually biased towards active neurons. Extracellular recordings by [44] that were unbiased towards responsive neurons suggest that many neurons have low spontaneous firing rates and that only a small fraction of neurons respond ‘well’ to stimuli in unanesthetized animals; this fact was also discussed in [43]. Moreover, calcium imaging experiments of awake and anesthetized rats in layer 2/3 of the cortex show that many neurons have resting firing rates less than 1 s−1 [45]. The actual firing rate of neurons in the resting state is a contentious issue, but our results hold for many parameter regimes (see Fig. 6).

Bottom Line: It has been shown that divisive gain modulation of neural responses can result from a stochastic shunting from balanced (mixed excitation and inhibition) background activity.However, input statistics, such as the firing rates of pre-synaptic neurons, are often dynamic, varying on timescales comparable to typical membrane time constants.Using a population density approach for integrate-and-fire neurons with dynamic and temporally rich inputs, we find that the same fluctuation-induced divisive gain modulation is operative for dynamic inputs driving nonequilibrium responses.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania, USA. chengly@math.pitt.edu

ABSTRACT
The modulation of the sensitivity, or gain, of neural responses to input is an important component of neural computation. It has been shown that divisive gain modulation of neural responses can result from a stochastic shunting from balanced (mixed excitation and inhibition) background activity. This gain control scheme was developed and explored with static inputs, where the membrane and spike train statistics were stationary in time. However, input statistics, such as the firing rates of pre-synaptic neurons, are often dynamic, varying on timescales comparable to typical membrane time constants. Using a population density approach for integrate-and-fire neurons with dynamic and temporally rich inputs, we find that the same fluctuation-induced divisive gain modulation is operative for dynamic inputs driving nonequilibrium responses. Moreover, the degree of divisive scaling of the dynamic response is quantitatively the same as the steady-state responses--thus, gain modulation via balanced conductance fluctuations generalizes in a straight-forward way to a dynamic setting.

Show MeSH
Related in: MedlinePlus