Noise-driven adaptation: in vitro and mathematical analysisReferencesNeurocomputing 52–54 (2003) 877 – 883www.elsevier.com/locate/neucomNoise-driven adaptation: in vitro andmathematical analysisLiam Paninski∗;1, Brian Lau, Alex ReyesCenter for Neural Science, New York University, 4 Washington Place, New York, NY 10003, USAAbstractVariance adaptation processes have recently been examined in cells of the /y visual system andvarious vertebrate preparations. To better understand the contributions of somatic mechanismsto this kind of adaptation, we recorded intracellularly in vitro from neurons of rat sensorimotorcortex. The cells were stimulated with a noise current whose standard deviation was variedparametrically. We observed systematic variance-dependent adaptation (de2ned as a scaling ofa nonlinear transfer function) similar in many respects to the e3ects observed in vivo. The factthat similar adaptive phenomena are seen in such di3erent preparations led us to investigate asimple model of stochastic stimulus-driven neural activity. The simplest such model, the leakyintegrate-and-2re (LIF) cell driven by noise current, permits us to analytically compute manyquantities relevant to our observations on adaptation. We show that the LIF model displays“adaptive” behavior which is quite similar to the e3ects observed in vivo and in vitro.c 2003 Elsevier Science B.V. All rights reserved.Keywords: Adaptation; Noise; Integrate-and-2re; Fokker–PlanckIt is widely understood that sensory neurons adapt to the prevailing statistics of theirinputs [10]. Fairhall et al. [5] recently reported one such adaptation process in the /yvisual system; they described a motion-sensitive neuron that appears to scale its input–output function to adapt its 2ring rate to the variance of the observed motion signal.However, the mechanisms underlying this type of contrast-dependent adaptation are un-known; speci2cally, it is unclear whether the observed phenomena arise from networkThis work was supported by NSF Grant IBN-0079619. LP and BL are supported by HHMI and NDSEGpredoctoral fellowships, respectively. We thank E. Simoncelli for many interesting discussions.∗Corresponding author.E-mail address: [email protected] (L. Paninski).1Contact: [email protected]; http://www.cns.nyu.edu/∼liam0925-2312/03/$ - see front matterc 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0925-2312(02)00819-6878 L. Paninski et al. / Neurocomputing 52–54 (2003) 877 – 883 2020 1 2 3 40102030Time (s)Rate (Hz) Current (nA)Fig. 1. Experimental details. Sagittal slices were prepared from adolescent and adult rats (P14-P24) asdescribed in [8]. Brie/y, slices were maintained at 30◦C in arti2cial cerebrospinal /uid consisting of (inmM): 125 NaCl, 2.5 KCl, 25 glucose, 25 NaHCO3, 1.25 NaH2PO4, 2 CaCl2, and 1 MgCl2. Cells werevisualized using infrared di3erential interference contrast microscopy with a 40× water immersion objective.Dual-electrode whole-cell recordings were made using pipettes with 5–15 M resistance when 2lled with(in mM): 100 K-gluconate, 20 KCl, 4 ATP-Mg, 10 phosphocreatine, 0.3 GTP, and 10 HEPES, pH 7.3(310 mOsm). Recordings were performed in current clamp using Axoclamp 2B ampli2ers (Axon Instruments,Foster City, CA), and stimulus presentation and data acquisition was managed using IGOR (Wavemetrics,Lake Oswego, OR). Gaussian white noise current stimuli were delivered through one electrode, while voltagewas recorded through the other electrode and processed on- and o3-line. Left panel shows a photograph ofa cell with the recording and stimulating electrodes partially visible; right panel shows a sample trace ofthe current input (including a jump between two values of noise variance), and the corresponding peri-eventtime histogram (note that the noise current was not “frozen,” that is, a new noise current was drawn i.i.d.for each trial).dynamics or from dendritic or somatic mechanisms in individual neurons. We hypoth-esized that (1) somatic mechanisms could account for at least part of the observedadaptation phenomena, and that (2) these somatic e3ects are general in the sense thatthey depend only weakly on the biophysical parameters governing a given neuron’sbehavior. To test hypothesis (1), we recorded intracellularly from layer V pyramidalneurons in sensorimotor cortex in vitro (see Fig. 1 for details), while stimulating witha noise current whose standard deviation (or “contrast”) was varied parametrically.Hypothesis (2) will be addressed mathematically below.For ease of comparison, we analyzed our data using the basic framework utilizedin [5] (see also, e.g., [4]). For each neuron, we estimated a separate spike-triggeredaverage (STA) at each current standard deviation [3], using data acquired after theneuron had reached a steady-state 2ring rate. We then projected our stimulus onto thenormalized STA function (this operation is equivalent to a time-reversed convolution,or 2ltering), and estimated, via a nonparametric histogram approach, the conditionalprobability of a spike given each observed value of the projected current. These con-ditional 2ring rate functions (termed N -functions, for nonlinearity, in keeping withconvention) will be the main object of our analysis (see Fig. 2 for an example).L. Paninski et al. / Neurocomputing 52–54 (2003) 877 – 883 8790 0.2 0.4 0.6 0.8 100.020.040.060.080.1P(Spike)Projected current (nA)1.2 nA0.6 nA0.3 nAσ =Fig. 2. Examples of N -functions for a single pyramidal neuron. Each curve represents data for a particularstandard deviation of input current.Our main observations are as follows: 2rst, the STA (which is often thought of as alinear pre2lter for the cell, the stimulus dimension to which the neuron is most sensi-tive) changed with the standard deviation of the injected current. As was increased,we observed a systematic reduction in the time-to-peak as well as the half-width ofthe STA, consistent with results seen in vitro [3] and in vivo [1]. We also observedchanges in the N -functions (Fig. 2). If we de2ne “gain” as the slope of the N -function,we have that the gain of the observed cortical cells was consistently inversely propor-tional to the standard deviation of the injected current; this result is strikingly similarto those of Fairhall et al. [5].What could explain the gain changes described above? One common model forgain changes in cortical cells requires the presence of some channel whose conduc-tance is