If neuron models such as the Hodgkin-Huxley or the integrate-and-fire model are driven by a sufficiently strong constant current, they generate a regular sequence of spikes. In neuronal models with adaptation currents5.1 there might be a short transient phase at the beginning, but then all interspike intervals are constant. Spike trains of typical neurons in vivo show a much more irregular behavior. Whether the irregularity is the sign of noise or of a rich code is at present an open question (Softky and Koch, 1993; Shadlen and Newsome, 1994; Bair and Koch, 1996). In the first subsection we review some evidence for neuronal variability and spike train irregularity. We then discuss potential sources of noise.
Many in vivo experiments show noisy behavior of central neurons. The activity of neurons from the visual cortex, for example, can be recorded while a slowly moving bar is presented on a screen within the visual field of the animal (Hubel and Wiesel, 1977,1959). As soon as the bar enters the neuron's receptive field the firing rate goes up until the bar leaves the receptive field at the opposite border. The spike train, however, varies considerably from trial to trial, if the same experiment is repeated several times. Furthermore, the very same neuron is spontaneously active even if the screen is blank and no external stimulus is applied. During spontaneous activity, the intervals between one spike and the next exhibit a large variability resulting in a broad distribution of interspike intervals; see e.g., Softky and Koch (1993).
Are these experiments convincing evidence for ubiquitous noise in the central nervous system? The above two observations refer to experiments on the neural system as a whole. The cortical neuron that is recorded from, does not only receive input from the retina, but also from many other neurons in the brain. The effective input to this neuron is basically unknown. It is thus possible that there is a substantial fluctuation in the input current to cortical neurons, even though the external (visual) stimulus is only slowly changing.
In fact, when neurons are driven by a known time-dependent intracellular input current, neurons seem to behave more or less deterministically (Zador, 1998; Bryant and Segundo, 1976; Mainen and Sejnowski, 1995). Moreover, if the external visual stimulus changes rapidly, neurons in the visual system react much more reliably than for constant or slowly moving stimuli (Maršálek et al., 1997; Bialek et al., 1991; de Ruyter van Steveninck et al., 1997; Berry et al., 1997; Bair and Koch, 1996). Whether a neuron behaves nearly deterministically or rather randomly thus depends, at least to a certain extent, on the stimulus.
We distinguish between intrinsic noise sources that generate stochastic behavior on the level of the neuronal dynamics; and extrinsic sources that arise from network effects and synaptic transmission (Manwani and Koch, 1999).
A source of noise, which is literally omnipresent, is thermal noise. Due to the discrete nature of electric charge carriers, the voltage u across any electrical resistor R fluctuates at finite temperature (Johnson noise). The variance of the fluctuations at rest is u2 R k T B where k is the Boltzmann constant, T the temperature and B the bandwidth of the system (Manwani and Koch, 1999). Since neuronal dynamics is described by an equivalent electrical circuit containing resistors (cf. Chapter 2.2), the neuronal membrane potential fluctuates as well. Fluctuations due to Johnson noise are, however, of minor importance compared to other noise sources in neurons (Manwani and Koch, 1999).
Another source of noise that is specific to neurons arises from the finite number of ion channels in a patch of neuronal membrane (White et al., 2000; Schneidman et al., 1998). Most ion channels have only two states: they are either open or closed. The electrical conductivity of a patch of membrane for ion type i is proportional to the number of open ion channels. For a given constant membrane potential u, a fraction Pi(u) of ion channel of type i is open on average. The actual number of open channels fluctuates around Ni Pi(u) where Ni is the total number of ion channels of type i in that patch of membrane. The formulation of the Hodgkin-Huxley equations in terms of ion channel conductivities (see Chapter 2.2) is implicitly based on the assumption of a large number of ion channels so that fluctuations can be neglected. Since, in reality, Ni is finite, the conductivity fluctuates and so does the potential. If the membrane potential is close to the threshold, channel noise can be critical for the generation of action potentials. Models that take the finite number of ion channels into account, can reproduce the observed variability of real neurons with intracellular stimulation (Chow and White, 1996; Schneidman et al., 1998). In particular, they show little spike jitter if the input current is rapidly changing, but are less reliable if the input current is constant.
Apart from intrinsic noise sources at the level of an individual neuron there are also sources of noise that are due to signal transmission and network effects (extrinsic noise). Synaptic transmission failures, for instance, seem to impose a substantial limitation to signal transmission within a neuronal network. Experiments with double electrode recordings from two synaptically connected neurons suggest that only 10-30 percent of presynaptic spikes generate a postsynaptic response (Markram and Tsodyks, 1996; Hessler et al., 1993).
Finally, an important part of the irregularity of neuronal spiking during spontaneous activity seems to be due to the properties of the network as a whole rather than to individual neurons. In model studies it has been shown that networks of excitatory and inhibitory neurons with fixed random connectivity can produce highly irregular spike trains - even in the absence of any source of noise (van Vreeswijk and Sompolinsky, 1996; Kistler and De Zeeuw, 2002; Brunel and Hakim, 1999; Nützel et al., 1994; Fusi et al., 2000). We will discuss the underlying mechanisms in Sections 6.4.3 and 8.3. As a result of the network activity, each neuron receives as input an irregular spike sequence that can be described as stochastic spike arrival; cf. Section 5.5. The difference between the large variability of neurons in vivo compared to that during intracellular stimulation in vitro can therefore be, at least partially, attributed to network effects.
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